2. Exponents & Roots

Question 1

What is the result of having a negative base?

Answer 1

When dealing with negative bases, pay particular attention to PEMDAS. Unless the negative sign is inside the parentheses, the exponent does not distribute
-2^4 = -16
(-2)^4 = 16

An exponential expression with negative base inside parentheses yields a positive number when the exponent is even and a negative number when the exponent is odd.
Examples:
(-1)^15 = -1
(-1)^16 = 1

Question 2

What happens with (fraction)^b?

Answer 2

When the base is a fraction between zero and one, the value decreases as the exponent increases. (1/2)^3 = 1/2 * 1/2 * 1/2 = 1/8, which is smaller than the starting fraction, 1/2

Question 3

What is the multiplication rule for exponents?

Answer 3

a^x * a^y = a^(x+y). When you multiply exponential terms that have the same base, add the exponents. To add exponents, you should see the same two bases

Question 4

What is the division rule for exponents?

Answer 4

a^x / a^y = a^(x-y). When you divide exponential terms that have the same base, subtract the exponents

Question 5

What is the successive power rule?

Answer 5

(a^x)^y = a^(xy). When you raise something that already has an exponent to another power, multiply the two exponents together. To multiply exponents, you should be applying two exponents, one after the other, to just one base

Question 6

What is the rule for negative exponents?

Answer 6

a^(-b) = 1 / a^b
OR
1 / a^-b = a^b

Something with a negative exponent is just “one over” the same thing with a positive exponent. This rule holds true even if the negative exponent appears in the denominator, or if the negative exponent applies to a fraction.

Question 7

What is the rule for applying an exponent to a product or fraction?

Answer 7

(ab)^x = a^x * b^x. When you apply an exponent to a product or fraction, apply the exponent to each factor on top and bottom. If you see two factors with the same exponent, then you might regroup the factors as a product

Question 8

What is the rule for adding/subtracting terms with the same or different bases?

Answer 8

13^5 + 13^3 = 13^3 * (13^2 + 1). When you add or subtract exponential terms, you look for a common factor and pull it out.

2^3 + 6^3 = 2^3 * (1 + 3^3). If you do not have the same bases in what you are adding or subtracting, then you cannot immediately factor. If the bases are numbers, break them down to small factors and see whether you now have anything in common

Question 9

What is the rule for fractional exponents?

Answer 9

If the exponent is a fraction, the numerator reflects what power to raise the base to, and the denominator reflects which root to take.

Example: 4^(2/3) = Cbrt(4^2).

Question 10

What are the basic rules of exponents?

Answer 10

(1) Multiplication: (a^b) * (a^c) = a^(b+c)
(2) Division: (a^b) / (a^c) = a^(b-c)
(3) Successive: (a^b)^c = a^(b*c)
(4) Negative Exponent: a^(-b) = 1 / a^b OR 1 / a^-b = a^b
(5) Factoring: (a-b)^2 = a^2 – 2ab + b^2
(6) Fractional Exponent: a^(b/c) = cRoot(a^b) OR cRoot(a)^b

Question 11

What should you do if you have two different bases that are numbers (e.g. 2^2 * 4^3 * 16)?

Answer 11

Try breaking the bases down to prime factors. You might discover that you express everything in terms of one base.

2^2 * 4^3 * 16 = 2^2 *2^6 *2^4 =2^12

Question 12

What is the difference between even and odd exponents?

Answer 12

-Even exponents hide the sign of the base, which means there can be up to two solutions
Example:
x^2 = 16
x = 4 OR x = -4

-Odd exponents keep the sign of the base, which means there can only be one solution
Example:
x^3 = -125
x = -5

Question 13

What are two examples of even exponent equations that do not yield two solutions?

Answer 13

*Only one solution
x^2 + 3 = 3
x = 0

*Squaring can never produce a negative number! Thus, there are no solutions to this equation
x^2 + 9 = 0
x^2 = -9

Question 14

How should you treat an equation that includes some variables with even exponents and some variables with odd exponents?

Answer 14

Treat it as dangerous! It is likely to have two solutions

Question 15

How do you approach problems that involve exponential expressions on both sides of the equation?

Answer 15

It is imperative to rewrite the bases so that either the same base or the same exponent appears on both sides of the exponential equation. Once you do this, you can usually eliminate the bases or the exponents and rewrite the remainder as an equation

(1) Rewrite the bases so that the same base appears on both sides of the equation
(2) Plug the rewritten bases into the original equation
(3) Simplify the equation using the rules of exponents
(4) Eliminate the identical bases, rewrite the exponents as an equation, and solve

*BEWARE: if 0, 1, or -1 is the base (or could be the base), the outcome of raising those bases to powers is not unique (e.g. 0^2 = 0^5 = 0; so if 0^x = 0^y, you cannot claim that x = y)

Question 16

Solve for w in the following equation: (4^w)^3 = 32^(w – 1)

Answer 16

(4^w)^3 = 32^(w – 1)
((2^2)^w)^3 = (2^5)^(w – 1)
2^(6w) = 2^(5w – 5)
6w = 5w – 5
w = -5

Question 17

Algebra Strategy Guide, Ch 3, Q 7. Simplify (4^y + 4^y + 4^y + 4^y)(3^y + 3^y + 3^y).

Answer 17

(4^y + 4^y + 4^y + 4^y)(3^y + 3^y + 3^y)
[4^(y + 1)][3^(y + 1)]
12^(y+1)

Question 18

Algebra Strategy Guide, Ch 3, Q 8. Solve for a in the following equation: 4^a + 4^(a+1) = 4^(a +2) – 176

Answer 18

The key to this problem is to express all of the exponential terms in terms of the greatest common factor of the terms, which is 4^a.

4^a + 4^(a+1) = 4^(a +2) – 176
176 = 4^(a +2) – 4^a –  4^(a+1) 
176 = 4^a * 4^2 – 4^a –  4^a * 4
176 = 4^a (4^2 – 1 –  4)
4^a = 176 / 11
4^a = 16
a = 2

Question 19

Algebra Strategy Guide, Ch 3, Q 9. If m and n are positive integers and (2^18)(5^m) = 20^n, what is the value of m?

Answer 19

With exponential equations such as this one, the key is to recognize that as long as the exponents are all integers, each side of the equation must have the same number of each type of prime factor

(2^18)(5^m) = 20^n
(2^18)(5^m) = (2 * 2 * 5)^n
(2^18)(5^m) = 2^n * 2^n * 5^n
2^18 * 5^m = 2^2n * 5^n
18 = 2n AND m = n
n = 9 = m

Question 20

What is a root?

Answer 20

The opposite of an exponent (in a sense). Sqrt(16) means: what number (or numbers), when multiplied by itself twice, will yield 16? In this case, both 4 and -4 would multiply to 16 mathematically. However, when the GMAT provides the square root sign for an even root, such as a square root, then the only accepted answer is the positive root, 4. That is, Sqrt(16) = 4, NOT +4 or -4. In contrast, the equation x^2 = 16 has TWO solutions, +4 and -4.

Question 21

What is the difference between even roots and odd roots?

Answer 21

A root using an even number, such as a square root or a fourth root, has only a positive value on the GMAT. NOTE: there is no solution for the root of a negative number (i.e. no number when multiplied an even number of times, can be negative)!

A root using an odd number, such as a cube root or a fifth root, will have the same sign as the base of the root. For example, CBRT(64) = 4, CBRT(-27) = -3.

Question 22

What happens when you square a square root or square-root a square?

Answer 22

Sqrt(x)^2 = x. If you square a square root, then you get the original number

Sqrt(x^2) = x If you square-root a square, then you get the absolute value of the original number

Question 23

Which is larger: Sqrt(x) or x?

Answer 23

The square root of a number that is larger than 1 is smaller than the original number, e.g. Sqrt(25) < 25.

However, the square root of a number between 1 and 0 is bigger than the original number, e.g. Sqrt(0.5) > 0.5

Question 24

What is the square root of -1, 0, and 1?

Answer 24

The square root of 1 is 1 and the square root of 0 is 0.

You cannot take the square root of a negative number (i.e. what is inside the radical sign must never be negative). Likewise the square root symbol never gives a negative result.

NOTE: When you see the square root symbol on the GMAT, only consider the positive root. In contrast, when YOU take the square root of both sides of an equation, you have to consider both the positive and negative roots (x^2 = 25, x could equal 5 or -5)

Question 25

To what is a square root equivalent?

Answer 25

If you take a square root of a positive number raised to a power, then rewrite the square root as an exponent of 1/2, then multiply exponents

e.g. Sqrt(7^22) = 7^11

Or rewrite what is inside the root as a product of two equal factors

e.g. Sqrt(5^12) = Sqrt(5^6 * 5^6)

Question 26

What is the fractional power rule?

Answer 26

If you raise a number to a fractional power, then you apply two exponents – the numerator as is and the denominator as a root, in either order

e.g. 125^(2/3) = [Cbrt(125)]^2

Question 27

What is the multiplication and division rule for square roots?

Answer 27

Sqrt(x) * Sqrt(y) = Sqrt(xy) OR Sqrt(x) / Sqrt(y) = Sqrt(x/y). When you multiply or divide square roots, multiply or divide the insides

Question 28

What do you do if you have a square root of a large number?

Answer 28

If you have the square root of a large number (or a root that does not match any answer choices), then you pull square factors out of the number under the radical sign

e.g. Sqrt(50) = Sqrt(25 * 2) = 5 * Sqrt(2)

If you do not spot a perfect square, you can always break the number down into primes. This method is longer but guaranteed.

Question 29

How do you determine the cube root of 216?

Answer 29

216 = 3 * 3 * 3 * 2 * 2 * 2 = 6 * 6 * 6 = 6^3

Question 30

What is the addition and subtraction rule for square roots?

Answer 30

If the operation between the terms is addition or subtraction, you cannot separate or combine the roots! Sqrt(4 + 9) DOES NOT EQUAL Sqrt(4) + Sqrt(9)

If you add or subtract underneath the square root symbol, factor out a square factor from the sum or difference. OR go ahead and crunch the numbers as written, if they are small.

Example: Sqrt(4^14 + 4^16) = Sqrt(4^14 * (1 + 4^2)) = 4^7 * Sqrt(17)

Question 31

What is an imperfect root versus a perfect root?

Answer 31

A perfect root yields an integer, e.g. Sqrt(16), whereas an imperfect root does not, e.g. Sqrt(52)

Question 32

How do you simplify an imperfect root?

Answer 32

Rewrite the root as the product of primes under a radical

Example: Sqrt(52) = Sqrt(2 * 2 * 13) = 2Sqrt(13)

Question 33

What is the conjugate and when is it used?

Answer 33

Conjugate: simply change the sign of the square root term. For (a + Sqrt(b)), the conjugate is (a – Sqrt(b)). For (a – Sqrt(b)), the conjugate is (a + Sqrt(b)).

*By multiplying the numerator and denominator by the conjugate, you can simplify a denominator that contains the sum or difference of a square root and another term

Question 34

What are the basic rules of roots?

Answer 34

(1) Square Roots: Sqrt(81) = 9 – there is only one positive answer when you are given the square root of something. X^2 = 81 – when YOU take the sqrt, there are two possible solutions
(2) Multiplication/Division: Sqrt(a * b) = Sqrt(a) * Sqrt(b) and Sqrt(a/b) = Sqrt(a) / Sqrt(b)
(3) Addition/Subtraction: Sqrt(a + b) does not equal Sqrt(a) + Sqrt(b)
(4) Root in Denominator: 3/Sqrt(x) = 3/Sqrt(x) * Sqrt(x)/Sqrt(x) = 3Sqrt(x) / x
(5) Conjugate: 3 / (5 + Sqrt(x)) = 3 / (5 + Sqrt(x)) * (5 - Sqrt(x)) / (5 - Sqrt(x)) = (15 - 3Sqrt(x)) / (25 – x)

Careful!
Sqrt(a + b) does not equal Sqrt(a) + Sqrt(b)
Sqrt(a - b) does not equal Sqrt(a) - Sqrt(b)
Sqrt(a^2 + b^2) does not equal a + b

Question 35

Simplify 4 / (3 – Sqrt(2))

Answer 35

4 / (3 – Sqrt(2)) * (3 + Sqrt(2)) / (3 + Sqrt(2))
= [12 + 4Sqrt(2)] / (9 – 7)
= [12 + 4Sqrt(2)] / 7

Question 36

Is (NEG)^POS positive or negative?

Answer 36

It can be either positive or negative depending on whether the exponent is even or odd

Odd exponent: negative value
Even exponent: positive value

Question 37

Ch 3, Q 50. What is the square root of 450?

Answer 37

Sqrt(2 * 3 * 3 * 5 * 5) = 15Sqrt(2)

Question 38

Ch 3, Q 53. What is the square root of 343?

Answer 38

Sqrt(7 * 7 * 7) = 7Sqrt(7)

Question 39

Recognition: 3^(n + 1)

Answer 39

(3^n ) * (3)

Question 40

Recognition: what does x^2 = 64 mean?

Answer 40

-8 < x < 8