GRE Math : Exponents and Rational Numbers

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Example Questions

Example Question #1 : How To Find An Exponent From A Rational Number

find x

8^x=2^x+6

Possible Answers:

2 or -1

-1

Correct answer:

Explanation:

8 = 2³

(2³)^x = 2^3x

2^3x = 2^x+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

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Example Question #2 : How To Find An Exponent From A Rational Number

Compare $3^{6}$ and $27^{2}$ .

Possible Answers:

$3^{6} > 27^{2}$

$3^{6} = 27^{2}$

$3^{6} < 27^{2}$

The relationship cannot be determined from the information given.

Correct answer:

$3^{6} = 27^{2}$

Explanation:

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

$27 = 3^{3}$

$27^2 = (3^{3})^2$

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

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

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Example Question #3 : Exponents And Rational Numbers

Solve for .

2^x=32

Possible Answers:

Correct answer:

Explanation:

can be written as 2^5.

Since there is a common base of , we can say

2^x=2^5 or x=5 .

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Example Question #3 : How To Find An Exponent From A Rational Number

Solve for .

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

Possible Answers:

$-\frac{1}{2}$

$\frac{1}{2}$

Correct answer:

Explanation:

The basees don't match.

However:

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

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

So, $(3^{-1})^{2}$ $=3^{-2}$ . x=-2 .

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Example Question #4 : How To Find An Exponent From A Rational Number

Solve for .

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

Possible Answers:

$\frac{1}{4}$

256

$-\frac{1}{4}$

Correct answer:

Explanation:

The bases don't match.

However:

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

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

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

x=-4

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Example Question #5 : How To Find An Exponent From A Rational Number

Solve for

$2^{x+1}=128$

Possible Answers:

Correct answer:

Explanation:

Recall that 128=2^7 .

With same base, we can write this equation:

x+1=7 .

By subtracting on both sides, x=6 .

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Example Question #2 : Exponents And Rational Numbers

Solve for .

$2^{x^2+4}=32$

Possible Answers:

-1, 1

Correct answer:

-1, 1

Explanation:

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

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

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

Take the square root of both sides we get BOTH and .

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Example Question #8 : Exponents And Rational Numbers

Solve for .

5^x=25^4

Possible Answers:

Correct answer:

Explanation:

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

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

$2\cdot 4=8$ .

x=8

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Example Question #9 : Exponents And Rational Numbers

Solve for .

$4^{2x}=16^6$

Possible Answers:

Correct answer:

Explanation:

There are two ways to go about this.

Method

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

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

Divide on both sides to get x=6 .

Method :

We can change the base from to 16.

$4^{2x}=(4^2)^x=16^x$

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

We have the same base so basically x=6 .

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Example Question #10 : Exponents And Rational Numbers

Solve for .

1024^x=2

Possible Answers:

$\frac{1}{10}$

$-\frac{1}{10}$

Correct answer:

$\frac{1}{10}$

Explanation:

Since we can write $1024^x=(2^{10})^x$ .

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

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

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GRE Math : Exponents and Rational Numbers

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #1 : How To Find An Exponent From A Rational Number

Example Question #2 : How To Find An Exponent From A Rational Number

Example Question #3 : Exponents And Rational Numbers

Example Question #3 : How To Find An Exponent From A Rational Number

Example Question #4 : How To Find An Exponent From A Rational Number

Example Question #5 : How To Find An Exponent From A Rational Number

Example Question #2 : Exponents And Rational Numbers

Example Question #8 : Exponents And Rational Numbers

Example Question #9 : Exponents And Rational Numbers

Example Question #10 : Exponents And Rational Numbers

All GRE Math Resources

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