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Archive for July, 2010

Quadrilaterals

By posted July 30, 2010

On the SAT and ACT, you’ll have to be familiar with your shapes. Unfortunately, this knowledge goes significantly beyond distinguishing a circle from a triangle, though you’ll still have to know that.

Let’s talk about a popular one: quadrilaterals. “Quadrilateral” is just a fancy, polysyllabic word  for any four-sided polygon. There are a few different types of quadrilaterals that you should be familiar, but, first, I’ll discuss some basic properties that all quadrilaterals have in common.

Universal Properties

  • All interior angles of a quadrilateral add up to 360 degrees.
  • Find the perimeter of any quadrilateral by adding up the four sides.

There are three basic types of quadrilaterals you will find on the test: parallelograms, rectangles, and squares.

Parallelograms

By definition, a parallelogram is any quadrilateral whose opposite sides are both parallel and equal in length. Notice in the diagram that the opposite angles are equal, and the consecutive angles–those  that share a side–are supplementary (they add up to 180 degrees).

Area of a Parallelogram: base times height, or bh. (Simpler than it looks)

Rectangles

Rectangles are special parallelograms in which all the angles are right angles. Note that they still retain the characteristic that opposite sides are equal.

Area of a Rectangle: base times height, or bh.

Squares

A square is a special type of rectangle in which all the sides are equal.

Area of a Square: base times height, or side^2

Special Information

Granted, the SAT and ACT will not simply test you on your ability to spot these special types of quadrilaterals. You’ll have to use your knowledge of their properties to perform measurements and calculations. Here are some familiar situations for quadrilateral measurement:

  1. Diagonals

You’ll often have to find the diagonal of a rectangle to yield new information about a shape. The diagonal is just the line that extends from one corner to another, and you can calculate a rectangle’s diagonal by using the Pythagorean Theroem: a² + b²= c². All rectangles are made up of two congruent right triangles. In fact, all squares are made up of two special congruent right triangles called 45-45-90 triangles. These special isosceles triangles are so named because they are made up of one 90 degree and two 45 degree angles. To save yourself time, remember this diagram:

With this information, you could find out the sides of a square (and therefore its area and perimeter) with just its diagonal measurement. For example, if the diagonal of a given square is 5, then I can use this equation to yield side n:

n*rad(2) = 5

n=5 / rad(2)

rationalize the denominator…

n=(5 *rad 2 ) / 2

  1. The Square in Circle Problem

You may see a square inscribed in a circle like the one above, and the question may ask you for the area of the square provided that the radius of the circle is 2.5. To solve this problem, remember that the diameter of this circle must be equal to the diagonal of the square–draw a line from two opposite corners and see for yourself. To solve it, just recognize that a radius of 2.5 yields a diameter of 5, which also happens to be the length of our diagonal. Use the steps above to find side n¸and simply square that n.

Those are the quadrilateral basics. Whenever you’re stumped on a word problem with measurements, remember the properties of quads and it may get you out of a jam!

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Imaginary and Complex Numbers

By posted July 29, 2010

You might be surprised that not all numbers are real–some are imaginary. No, imaginary numbers aren’t as interesting as you might imagine them to be. They’re merely numbers invented by mathematicians to signify the even roots of negative numbers. Yup, just when you thought the test-writers packed in enough math material for a standardized test, they incorporated a whole set of numbers that doesn’t correspond to anything in reality.

Imaginary Numbers

Imaginary numbers, represented by the letter i, represent the even roots of negative numbers. The square root of -1, for example, is i. If you never took Algebra 2, or you slept through the portion on imaginary numbers, you might still think that the square root of any negative number is mathematically impossible, or undefined (like 1/0). Well, in the world of real numbers, it is. That’s why a bunch of bored mathematicians invented imaginary numbers.

Let’s look at a couple of examples of how imaginary numbers work. As long as you remember that the definition of i is √(-1), you should be fine.

√(-16)=  √ (16) * √ (-1) = 4i

See? It’s that simple. Just tack on that little i to the roots of negative numbers.

Complex Numbers

A complex number is what we call the sum of a real number and an imaginary number. Think of it as a marriage of the real and imaginary, a tasty cocktail of Morpheus’s proffered red and blue pills. Complex numbers are written in the form a+bi, where a and b are real numbers; for example, 6+7i, is a complex number.

  1. 1. The powers of i

To work with complex numbers, you must remember the pattern of the powers of i. Luckily, the pattern works in cycles of four:

i ^1= i

i ^2=-1

i^ 3=–i

i ^4=1

It’s much easier to simply remember the pattern than to work out the powers as products of √(-1).

By knowing the pattern, you can easily figure out a much larger exponent, say i^99. To figure this out, think of the closest multiple of 4 that’s less than the exponent; in this case, it’s 96. So, i^99 is the same thing as i^(96+3), which means that the corresponding exponent in the pattern is 3 (96=exponent of 4, 97=exponent of 1, 98=exponent of 2 , 99=exponent of 3, 100=exponent of 4, and so on). With an exponent of 3, i^99 must be -i.

  1. Operations

Operations on complex numbers is virtually identical to simplifying or expanding real numbers with variables; the only difference is that you must remember to apply the exponent rule whenever necessary.

Example: Expand (2x+i)(4x+3i)

First, we just use a basic FOIL method to expand:

8x^2 +6xi +4xi + 3i^2

8x^2 +10xi + 3i^2

Notice the i squared, and remember the pattern.

8x^2 +10xi + 3 (-1)

8x^2 +10xi -3

The most important thing to remember about imaginary numbers is the pattern of exponents. For the most part, dealing with imaginary numbers is pretty similar to dealing with polynomials (though do not mistake i for just another variable–it hates that). Just think of complex numbers as polynomials with a new set of rules to follow, and you’ll be fine.

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Identifying Sentence Errors

By posted July 27, 2010

On the SAT Writing section, there will be 18 total Identifying Sentence Error questions, or ISE’s. They count for the largest percentage of your Writing score. You’ll soon see how a firm understanding of grammar and a confident process of elimination are all you need to get most of these questions correct!

1. Check each underlined portion individually. What part of speech is underlined? Is it a verb, preposition, adjective, adverb, pronoun, etc? You’ll want to make sure you know your parts of speech. The SAT loves to test the same errors over and over, so knowing each part of speech will be a big clue. Here are some of the most common errors:

-          Verb – check the Subject-Verb agreement, Verb tense, number, etc.

-          Pronoun – check for a clear Antecedent, does the Pronoun agree with the noun in number?

-          Preposition – is the transition appropriate? Is it idiomatically correct? Make a flashcard of the most common  Idioms and learn them like you would vocabulary words. Idioms alone account for approximately 10% of your SAT Writing score!

-          Adverb/Adjective – remember that adjectives can only describe nouns, while adverbs can describe verbs, adjectives and other adverbs. Is there a word that is modifying a verb that needs an –ly suffix?

Incorrect: She moved quick.

Correct: She moved quickly.

The more you practice, the more you will see that certain errors occur commonly with certain parts of speech.

2. Check for style errors. After checking for errors within the underlined portions, go back and re-read the sentence as a whole. Specifically look for errors related to Parallelism, Comparisons, Transitions, and Wordiness. These errors will muddle the clarity and meaning of the sentence. Does the meaning of the sentence make sense to you as written?

3. Trust yourself. After you’ve aggressively identified each underlined part of speech and checked for style errors, you may not be able to find an error. Don’t second-guess yourself. You’re probably right! It’s important to remember that 5-8 of them will have “No Error”. That is somewhere around 1/3 of all ISE’s!

As you study, if you find yourself choosing (E) too often, you probably need to spend more time studying the most common types of grammatical SAT

Start your grammar practice by reading these other articles about Identifying Sentence Errors here.

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Inequalities

By posted July 26, 2010

We can break inequalities questions on the ACT down into three types: those involving a word problem, those involving algebra and those involving absolute values. Let’s tackle the word problems first because that generally involves translating the word problem rather than any algebraic calculations.

Translating Word Problems

Suppose a pulley can handle no more than 800 lbs of weight. It is currently holding 4 steel frames that weigh 112 lbs each. Amanda wants to load as many bricks onto it as she can without it breaking. If x represents the total weight of bricks, in lbs, that she can add, which of the following inequalities could be used to determine possible values of x?

From the question, I have deduced the following information:
• The pulley can handle no more than 800lbs. So, TOTAL WEIGHT ≤ 800
• I have 4 frames, each weighing 112 lbs. That means, 4 FRAMES WEIGH 4*112 = 448 lbs
• In addition to the 4 frames, I want to load x lbs in bricks.
I can thus conclude that TOTAL WEIGHT = 4 FRAMES + x = 448 + x and this must be  ≤ 800
In mathematical notation, x + 4(112) < 800 Don’t get tricked if the answer is written as 800 > x + 4(112). This statement is exactly the same as the statement above. 800 is greater than x + 4(112) is the same as x + 4(112) is less than 800.

Algebraic Inequalities

Another type of inequality involves algebra.  Suppose y = 2x + 4 and x < 3 and you need to find an inequality involving y.

  1. Start with the inequality you have x < 3
  2. Look at the equation involving y.  There is a 2x in it.  Since x < 3, that means 2x < 6
  3. If 2x < 6, that means 2x + 4 < 6 + 4 = 10.  (because we added 4 to each side)  Thus 2x + 4 < 10
  4. But 2x + 4 is simply y, so we can conclude that y < 10

The trick is to make your inequality look like the equation.  Can you work the next two examples out on your own?

If 2y = 2x + 4 and x < 3, find an inequality involving y

Answer: Like before, we have 2x < 6 and 2x + 4 < 10.  But now we have 2y < 10, meaning y < 5

If y = -2x + 4 and x < 3, find an inequality involving y

Hint: Don’t forget, that when multiplying an inequality by a negative number, you have to switch the signs.

Answer: If x < 3 , then -2x > -6.  That means that -2x + 4 > -2 so we get y > -2

Absolute Value Problems

The last type of question tests your knowledge of absolute values.  Absolute values are denoted by two straight lines | |.  Absolute values make negative numbers positive.

So |10| = 10 and |-10| = 10 too.

Now that we’ve established what absolute values do, we can solve absolute value inequalities.  Here’s a Grockit question:

A theater company is auditioning for actors to portray the leading character in a new play. The company is looking for actors between the ages of 20 and 40 (inclusive). Which of the following inequalities can be used to determine whether an actor of age a is eligible to audition for the part?

A)     | a-10 | ≤ 40

B)     | a-20 | ≤ 40

C)     | a-30 | ≤ 20

D)     | a-30 | ≤ 10

E)      | a-35 | ≤ 5

The answer is 20 ≤ a ≤ 40. But how do we make that look like one of the inequalities above?

  1. Take the average of 20 and 40.  That’s 30.
  2. Subtract 30 from everything to get 20 – 10 ≤ a – 30 ≤ 40 – 30
  3. You get -10 ≤ a – 30 ≤ 10  Note that the left and right side of the inequality is the same number, except that the one of the left is negative
  4. Now we can say, | a – 30 | ≤ 10  This basically means that a falls within 10 units more or less of 30, which is true since a ranges between 20 and 40.

So the answer is clearly choice D.

Can you work out the lower and upper bounds of the other choices?   I’ll do Choice A for you.

Choice A says | a – 10 | ≤ 40.

=> -40 ≤ a – 10 ≤ 40

=> -40 + 10 ≤  a – 10 + 10 ≤ 40 + 10

Thus Choice A is effectively saying -30 ≤ a ≤ 50, which is not the answer at all.

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ACT MC: Numbers and Operations

By posted July 22, 2010

Numbers and operations is the area in ACT Math used in most of the word problems and problems involving percentages, averages, and sequences.

Number Classifications
All numbers can be put into one or more of the following classes. In the ACT Math you will only deal in whole numbers and fractions. While all fractions can be represented in decimal form, it is generally advisable to keep them as fractions. All multiple choice selections will be presented in either fraction or whole number form.

Integers: are whole numbers. All integers greater than zero are known as positive integers, and all integers less than zero are known as negative integers. Zero is neither positive nor negative.

Consecutive Integers: are integers in sequence, without skipping any integers. An example of this is the series {9,10,11,12,13,14}. A consecutive integer series may also include zero, for example the series {-2,-1,0,1,2,3).

Odd Integers: are all of the integers that have a remainder 1 when divided by 2. So, {1,3,5,7,9,11,13,…}. The negatives of any of these numbers is also odd, so another example of a set of odd numbers is {…,-3,-1,1,3,5,…}.

Even Integers: are all integers that are divisible by 2 with no remainder. Zero is considered an even number, despite the fact that it is neither positive nor negative and not divisible by 2.

Some Rules Regarding even and odd numbers:
even ± even = even
even ± odd = odd
odd ± odd = even
even × even = even
even × odd = even
odd × odd = odd

Prime Numbers: are numbers that are divisible with no remainder by only 1 and itself. By definition 1 is not considered a prime number, the smallest prime number is 2. The set of prime numbers includes {2,3,5,7,11,13,17,…}  It is important to know how to factorize non prime numbers into prime numbers, so if you’re uncertain, you should read our post on it.

Rational Numbers: are numbers that can be expressed as a fraction of integers. For example 0.25 is rational because it can be expressed as ¼ and 1 and 4 are integers. A non-rational number will have a never-ending decimal form because it cannot be divided nicely.  All numbers which do not fit these criteria are called irrational numbers. An example of an irrational number is π.

Averages
The average of a set of numbers is the sum of that set of numbers divided by the number of elements in the set. So if you have the set of numbers {3,7,4,9,2,3}, there are 6 numbers in the set and the average is
average=(3+8+5+9+2+3)/6=30/6=5
The average does not necessarily have to be part of the set of numbers, although in our example it was.

Percent Increase and Decrease:
Percent is hundredths, which means out of a hundred.

For example
60%= 60/100=3/5=0.6
All percentages can be represented as a number with a % sign after it, a fraction, or a decimal.
Percent increase means the percent of the original that a value increases. Percent decrease means the percent of the original that a value decreases. The definition is

% increase=increase/original
% decrease=decrease/original
For example, a 40% increase on 20 is
40%= 4/10=increase/20
increase=80/10=8
Since the increase is 8, the new total will be 28.

Average Speed
The average speed of a time period is defined as the total distance traveled divided by the time it takes to travel that distance.
average speed=(distance traveled)/time

If you are dealing with a situation where you travel at two different speeds for different amounts of time, you can calculate the total distance traveled by multiplying the time of each leg with its respective speed. So if you travel x hours at a miles per hour, and y hours at b miles per hour, the total distance traveled is represented by
total distance traveled=x hr∙((a mi)/hr)+y hr∙((b mi)/hr)

From there, you can find the average speed of the entire trip by dividing the total distance traveled by the amount of time it took to travel that entire distance. In this case, the total time required was x+y, so
average speed=(total distance traveled)/(x+y)

Sequences
Sequences are sets of numbers where each number is found by performing arithmetic operations involving the term or terms preceding it. Two common sequences are arithmetic and geometric sequences.

Arithmetic Sequences: are sequences where you get the next number by adding some constant to the number before it.  For example, {3,9,15,21,27,…} is an arithmetic sequence because each term is 6 greater than the term before it.

Geometric Sequences: are sequences where you get the next number by multiplying the previous number by some constant.  For example, {4,12,36,108,…} is a geometric sequence because each term is equal to the term before it multiplied by 3.

Sequences are not limited to arithmetic and geometric. Each term can have any relation to the term before it, be it addition, subtraction, multiplication or division. A term may also be defined by multiple terms preceding it.  For example, 1, -1, 3, -3, 5, -5, 7, -7 is a sequence of odd terms where it changes from being positive to negative.

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ACT English: Conjunction Junction, What’s Your Function?

By posted July 21, 2010

You may not remember the song from Schoolhouse Rock (google it!) but you’ll definitely want to know your coordinating conjunctions and how they are used on the ACT.

The easiest mnemonic device to remember conjunctions is called FANBOYS, an acronym formed by the first letter of each conjunction:

For

And

Nor

But

Or

Yet

So

These conjunctions are called coordinating conjunctions because they join similar ideas.

Here’s an ACT English practice question. Good luck!

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Sequences on the SAT

By posted July 20, 2010

There are four main categories of sequences that appear on the SAT math sections. Think of sequences as a simple pattern, and detecting this pattern is probably the most difficult part.

Arithmetic Sequences

An Arithmetic Sequence is when subsequent terms in a sequence increase (or decrease) by a constant amount. Here’s the standard formula:

  • a_n = a_1 + (n – 1)d, where a_n is the value at term n, and d is the constant change.

Note that a_n – a_(n-1) = d

If given the following sequence, we can derive both a_1and d, to solve for any term.

8, 11, 14, 17, 20, 23…..    (a_1 = 8 and d = 3)

So, if asked what term number 86 is, we can just plug in to the formula:

= a_1 + (n – 1)d

a_n = 8 + (86 – 1)3 = 8 + 85*3 = 8 + 255 = 263

The problem with sequences though is that we all learned it differently.  Some of us learn it by starting on the “zeroth” term, some of us start on the “first term”.  Whatever the case, it will affect your general formula, so make sure you know what values “n” starts from.  In our case, n starts from 1, the first term.

Try this SAT practice question and test your math skills.

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Verb Tense

By posted July 19, 2010

Verb tense is often a simple error to spot on English multiple choice problems, but when it comes to harder problems, you really should know the ins and outs of verb tense.

It’s easy to spot errors like “I will play basketball yesterday.” We know that such a sentence is logically impossible, and to fix it, we simply change the future tense to the past tense: “I played basketball yesterday.”

Many problems, however, aren’t so simple. To master these, it’s best to learn the different tenses in English–the ones most often tested on the ACT–and their functions.

First, let’s take a look at the simple tenses:
Simple Present: The man runs. (He’s running right now)
Simple Past: The man ran yesterday. (He ran in the past)
Simple Future: The man will run tomorrow. (He will run in the future)

You may be thinking “duh,” but it helps to organize your knowledge of tenses–even the easy ones.

Here are the perfect tenses. The perfect tenses are a bit more complicated.

Present Perfect: I have practiced, so I am ready for the recital.

The present perfect often indicates something that you have just done, or something that you did in the past and may continue to do or are doing at the moment. Think of this example where we use present perfect vs. past: John (worked / has worked) at the coffee shop for three years. Notice that “worked” would tell us that John no longer works at the coffee shop; he’s talking about three years of his life that have already passed. If we use “has worked,” though, it suggests that John has worked at the shop for three years and continues to work there.

Past Perfect: I had practiced, so I played well at the recital.

The past perfect is used to indicate an action that occurred before another action in the past. Notice that both the practicing and the recital took place in the past, but we want to communicate the order of events: practicing took place before the recital. Thus, we use the past perfect for the practicing, and the simple past for the “playing well.”

Future Perfect: I will have practiced, so I will play well in the recital.

The future perfect is used to indicate an action that took place before another action in the future. We know that the practicing and the recital will both take place in the future; I will certainly practice before the recital but haven’t done so yet. This example sentence suggests that I have made a prediction about events in the future. It’s like saying: “Don’t worry about me. By the time the recital comes around, I will have practiced, and I will be great.”

Now that we know the tenses, let’s explore what some tense problems might look like.

Here’s an example from Grockit to get us started:

Without a doubt, one of the most interesting things about our trip to Paris next May was the change from speaking in English to speaking in French.

Here, we have to figure out the true tense of the sentence from a tense cue, “Our trip to Paris next May.” If the trip happens next May, then it will take place in the future. Thus, we cannot speak of the trip in past tense with the verb “was,” so you need to change that “was” to “will be.”

Here’s another example that tests something different while still testing knowledge of tenses:

When he claimed that he had spoke to the dignitary, Ken neglected to mention that the correspondence had been conducted chiefly through her secretary.

The problem here is not with the kind of tense used, but the improper application of that tense. The author wants to use the past perfect of “speak,” but says “had spoke” instead of “had spoken.” In all the perfect tenses, we must use the past participle of the word, which does not always look like the past tense of a word. The past participle of speak is “spoken.” The past participle of drink is “drunk.” The past participle of swim is “swum.” If any of these surprise you, review a list of irregular English verbs to fortify yourself against these nitpicky tense questions.

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SAT Math: Domain and Range

By posted July 16, 2010

The “domain” and “range” of a function are just some fancy terms for concepts you are already quite familiar with: the x values and the y values, respectively. The domain refers to the set of possible inputs (a.k.a. x values) of any given function, while the range refers to the set of possible outputs. You could say that for the given function f(x) = 2x+3, both the domain and range consist of “all real numbers” because any value of x can produce any result y. But, for a function like f(x)= 1/x, x cannot be zero (since 1/0 is like, well, impossible!), so our domain for that function is x<0 and x>0. And, what about a function like h(x) =√x;  since a negative number has no square root, h(x) has a domain of x>0.

1. Finding the Domain

Just like we did in the examples above, finding the domain of any given function is all about identifying restrictions on possible x values. These restrictions can be either…

a. Division by zero:  Since dividing by zero is undefined, or impossible, watch out for rational expressions that might yield zero in the denominator. For example, in the function     f(x)= 1 / 2x+6, x cannot be -3, since that would yield a zero in the denominator.

b. Negatives in square roots: This one is pretty self-explanatory. You cannot take the square root of a negative number (well, mathematicians have devised imaginary numbers for this very purpose, but they will not show up on the SAT general test).

Granted, finding the these restrictions will be a bit more complicated than simply spotting a variable under square root or in the denominator–it may take some good ol’ fashioned algebra.

In this case, all I have to do is factor the denominator, yielding (x+2)(x+3)=0. Therefore, x is zero at -2 and -3, and my domain consists of all real numbers except for -2 and -3. You may see this written as {x: x ≠ –2, –3}.

There are a couple things to watch out for here. Most obviously, x cannot be 7, since that would yield 0 in the denominator. Secondly, x cannot be greater than 4, since that would yield a negative number under a square root. The domain, then, is the set of real numbers such that x≥4,   x ≠ 7.

2. Finding the Range

Just like there are restrictions on the domain, there restrictions on the range, or y values.

a. Absolute Value: Since the absolute value of any number x is always positive, a function f(x)=|x| has a range of all real numbers greater than or equal to 0.

b. Even exponents: Since raising a negative number to an even exponent results in only positive numbers, expect variables with even exponents to yield a range above the x axis.

In this simple example, notice that the absolute value in the numerator renders the range positive values only.

Example 4: Find the range of

To solve this one, start close and work your way outward; each successive restriction–absolute value, the square root, etc–will further modify the range.

First, the absolute value renders the range positive values only, or  0 ≤ f(x) ≤ ∞.

Next,  the 4 under the square root shifts our range four units up, so our lower bound is no longer 0, but 4: 4 ≤f(x) ≤ ∞.

Next, noticing that the 4 is underneath the square root changes that lower bound to a 2:   2 ≤f(x) ≤ ∞.

Finally, the 2 in the denominator of the expression makes us divide our lower bound by 2, giving us 1 ≤ f(x) ≤ ∞.

Those are the basics of finding domain and range. To practice this skill, try to find the domain and range of any function you run into, like practice problems on Grockit .

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SAT Math: Special Right Triangles

By posted July 15, 2010

While you are given the ratios for special right triangles in the front of your SAT test booklet, knowing them now will save you precious time on test day! It can be especially efficient to memorize some common Pythagorean triplets.

Rather than use the Pythagorean theorem to find the third side of a right triangle every single time, you may notice that you often encounter right triangles with the ratios of 3:4:5 and 5:12:13. These ratios will also be true for any multiples of 3:4:5 and 5:12:13 such as 6:8:10 or 10:24:26.

For this triangle to the left, even without using Pythagorean, we know the third side must be 5.

There are also two right triangles that are very important to know called the special right triangles. These are so called because the ratio of their sides never changes. The first is a 30-60-90 triangle. Its sides will always be in a ratio of x: x√3 : 2x.

The other special triangle is the 45-45-90 triangle. Its sides will always be in a ratio of x: x: x√2.

It’s important to remember that for the 30-60-90 triangle, the hypotenuse is the side that has the ratio of 2x. Don’t confuse it with the 45-45-90 ratio, and think that the x√3 should be there!

Now let’s look at a sample Grockit problem:

If triangle ABC is a 30°-60°-90° right triangle, which of the following sets could represent triangle ABC’s side lengths?

A          2, 2, 2

B          2, 2, 2√2

C         2, 2√2, 2√2

D         2, 2√2, 2√3

E          2, 2√3, 4

For each answer choice x = 2, so knowing that the ratio of a 30-60-90 is x: x√3 : 2x, we can plug x in to get: 2: 2√3 : 2(2) or 2: 2√3 : 4. The answer is E.

Now let’s look at an example with the 45-45-90 triangle:

Which of the following sets of three numbers could be the side lengths, in yards, of a right triangle containing a 45° angle?

A          1, 1, 1

B          1, 21/2, 21/2

C         2, 2, 2(21/2)

D         1, 21/2, 31/2

E          1, 31/2, 2

It’s important to recognize that a fractional exponent is just another way of expressing a root.

21/2 = √2. Remember your exponent rules! We know the ratio for a 45-45-90 is x: x: x√2, which means two of the sides must be equal. That eliminates D and E. Out of the remaining choices, only C correctly expresses the ratio.

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