GRE Math : Exponents and Rational Numbers

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #21 : Algebra

 

find x

8x=2x+6

Possible Answers:

3

2 or -1

-1

4

2

Correct answer:

3

Explanation:

8 = 23

(23)x = 23x

 

23x = 2x+6  <- when the bases are the same, you can set the exponents equal to each other and solve for x

3x=x+6

2x=6

x=3

Example Question #1 : Exponents And Rational Numbers

Compare 3^{6}\displaystyle 3^{6} and 27^{2}\displaystyle 27^{2}.

Possible Answers:

3^{6} = 27^{2}\displaystyle 3^{6} = 27^{2}

3^{6} < 27^{2}\displaystyle 3^{6} < 27^{2}

3^{6} > 27^{2}\displaystyle 3^{6} > 27^{2}

The relationship cannot be determined from the information given.

Correct answer:

3^{6} = 27^{2}\displaystyle 3^{6} = 27^{2}

Explanation:

First rewrite the two expressions so that they have the same base, and then compare their exponents.

27 = 3^{3}\displaystyle 27 = 3^{3}   

27^2 = (3^{3})^2\displaystyle 27^2 = (3^{3})^2

Combine exponents by multiplying: (3^{3})^2 = 3^6\displaystyle (3^{3})^2 = 3^6

This is the same as the first given expression, so the two expressions are equal.

Example Question #3 : Exponents And Rational Numbers

Solve for \displaystyle x

\displaystyle 2^x=32

Possible Answers:

\displaystyle 4

\displaystyle 8

\displaystyle 6

\displaystyle 7

\displaystyle 5

Correct answer:

\displaystyle 5

Explanation:

\displaystyle 32 can be written as \displaystyle 2^5. 

Since there is a common base of \displaystyle 2, we can say

 \displaystyle 2^x=2^5 or \displaystyle x=5

Example Question #25 : Algebra

Solve for \displaystyle x.

\displaystyle 3^x=\frac{1}{9}

 

Possible Answers:

\displaystyle 3

\displaystyle -\frac{1}{2}

\displaystyle 2

\displaystyle \frac{1}{2}

\displaystyle -2

Correct answer:

\displaystyle -2

Explanation:

The basees don't match.

However: 

\displaystyle \frac{1}{3}=3^{-1} thus we can rewrite the expression as \displaystyle \frac{1}{9}=\left(\frac{1}{3}\right)^2.

Anything raised to negative power means \displaystyle 1 over the base raised to the postive exponent. 

So, \displaystyle (3^{-1})^{2}\displaystyle =3^{-2}\displaystyle x=-2

Example Question #2 : Exponents And Rational Numbers

Solve for \displaystyle x.

\displaystyle \frac{1}{4}^x=256

Possible Answers:

\displaystyle \frac{1}{4}

\displaystyle -4

\displaystyle 4

\displaystyle 256

\displaystyle -\frac{1}{4}

Correct answer:

\displaystyle -4

Explanation:

The bases don't match.

However: 

\displaystyle 4=\frac{1}{4}^{-1} and we recognize that \displaystyle 256=4^4.

Anything raised to negative power means \displaystyle 1 over the base raised to the postive exponent. 

\displaystyle \left(\frac{1}{4}^{-1}\right)^4=\frac{1}{4}^{-4}.  

 \displaystyle x=-4

Example Question #31 : Algebra

Solve for \displaystyle x.

\displaystyle 2^{x+1}=128

Possible Answers:

\displaystyle 5

\displaystyle 9

\displaystyle 6

\displaystyle 8

\displaystyle 7

Correct answer:

\displaystyle 6

Explanation:

Recall that \displaystyle 128=2^7

With same base, we can write this equation: 

\displaystyle x+1=7

By subtracting \displaystyle 1 on both sides, \displaystyle x=6

 

Example Question #6 : How To Find An Exponent From A Rational Number

Solve for \displaystyle x.

\displaystyle 2^{x^2+4}=32

Possible Answers:

\displaystyle -1, 1

\displaystyle -1

\displaystyle 1

\displaystyle -5

\displaystyle 5

Correct answer:

\displaystyle -1, 1

Explanation:

Since \displaystyle 32=2^5 we can rewrite the expression.

With same base, let's set up an equation of \displaystyle x^2+4=5.

By subtracting \displaystyle 4 on both sides, we get \displaystyle x^2=1.

Take the square root of both sides we get BOTH \displaystyle 1 and \displaystyle -1

Example Question #3 : Exponents And Rational Numbers

Solve for \displaystyle x.

\displaystyle 5^x=25^4

Possible Answers:

\displaystyle 10

\displaystyle 5

\displaystyle 6

\displaystyle 4

\displaystyle 8

Correct answer:

\displaystyle 8

Explanation:

They don't have the same base, however: \displaystyle 25=5^2.

Then \displaystyle 25^4=(5^2)^4. You would multiply the \displaystyle 2 and the \displaystyle 4 instead of adding.

\displaystyle 2\cdot 4=8

\displaystyle x=8

Example Question #4 : Exponents And Rational Numbers

Solve for \displaystyle x.

\displaystyle 4^{2x}=16^6

Possible Answers:

\displaystyle 12

\displaystyle 8

\displaystyle 6

\displaystyle 4

\displaystyle 10

Correct answer:

\displaystyle 6

Explanation:

There are two ways to go about this.

Method \displaystyle 1:

They don't have the same bases however: \displaystyle 16=4^2. Then \displaystyle 16^6=(4^2)^6

You would multiply the \displaystyle 2 and the \displaystyle 6 instead of adding. We have \displaystyle 2\cdot 6=2x

Divide \displaystyle 2 on both sides to get \displaystyle x=6.

 

Method \displaystyle 2:

We can change the base from \displaystyle 4 to \displaystyle 16.

\displaystyle 4^{2x}=(4^2)^x=16^x 

This is the basic property of the product of power exponents. 

We have the same base so basically \displaystyle x=6

Example Question #8 : Exponents And Rational Numbers

Solve for \displaystyle x.

\displaystyle 1024^x=2

Possible Answers:

\displaystyle -\frac{1}{10}

\displaystyle 2

\displaystyle \frac{1}{10}

\displaystyle 10

\displaystyle -10

Correct answer:

\displaystyle \frac{1}{10}

Explanation:

Since we can write \displaystyle 1024^x=(2^{10})^x

With same base we can set up an equation of \displaystyle 10x=1 

Divide both sides by \displaystyle 10 and we get \displaystyle x=\frac{1}{10}

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