How to find an exponent from a rational number

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GRE Quantitative Reasoning › How to find an exponent from a rational number

Questions 1 - 10

1

Solve for .

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

$\frac{1}{4}$

$-\frac{1}{4}$

Explanation

The bases don't match.

However:

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

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

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

2

Solve for .

$\frac{1}{10}$

$-\frac{1}{10}$

Explanation

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

With same base we can set up an equation of

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

3

Solve for .

Explanation

can be written as

Since there is a common base of , we can say

or .

4

Solve for .

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

$\frac{1}{2}$

$-\frac{1}{2}$

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

5

Solve for

$2^{x+1}=128$

Explanation

Recall that .

With same base, we can write this equation:

.

By subtracting on both sides, .

6

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

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

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

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

The relationship cannot be determined from the information given.

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.

7

Solve for .

$1024^x=\frac{1}{2}$

$-\frac{1}{10}$

$\frac{1}{10}$

Explanation

$1024^x=(2^{10})^x$

We still don't have the same base however: $2=\frac{1}{2}^{-1}$

Then,

$1024^x=(2^{10})^x=\left[\left(\frac{1}{2}\right)^{-1}\right]^{10x}$ .

With same base we can set up an equation of .

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

8

Solve for .

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

Explanation

Since we can rewrite the expression.

With same base, let's set up an equation of .

By subtracting on both sides, we get .

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

9

Solve for .

Explanation

They don't have the same base, however: .

Then . You would multiply the and the instead of adding.

$2\cdot 4=8$ .

10

Solve for .

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

Explanation

There are two ways to go about this.

Method

They don't have the same bases however: . Then

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

Divide on both sides to get .

Method :

We can change the base from to

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

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

We have the same base so basically .

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