Exponents and Rational Numbers - GRE Quantitative Reasoning

Card 0 of 16

Question

Quantity A: $(-1)^{137}$

Quantity B:

Answer

(–1) 137= –1

–1 < 0

(–1) odd # always equals –1.

(–1) even # always equals +1.

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Question

Quantitative Comparison: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given.

Quantity A Quantity B

43 34

Answer

In order to determine the relationship between the quantities, solve each quantity.

43 is 4 * 4 * 4 = 64

34 is 3 * 3 * 3 * 3 = 81

Therefore, Quantity B is greater.

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Question

$2^{-5}$

Answer

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

$2^{-5}=\frac{1}{2^5}=\frac{1}{32}$

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Question

Simplify

$2^{10}+2^9$

Answer

Whenever you see lots of multiplication (e.g. exponents, which are notation for repetitive multiplication) separated by addition or subtraction, a common way to transform the expression is to factor out common terms on either side of the + or - sign. That allows you to create more multiplication, which is helpful in reducing fractions or in reducing the addition/subtraction to numbers you can quickly calculate by hand as you'll see here.

So let's factor a .

We have .

And you'll see that the addition inside parentheses becomes quite manageable, leading to the final answer of $2^9\cdot 3$ .

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Question

Which of the following is not the same as the others?

Answer

Let's all convert the bases to .

$4^{12}=[2^2]^{12}=2^{24}$

$64^4=(2^6)^4=2^{24}$

$16^8=[2^4]^8=2^{32}$

$\left(\frac{1}{2}\right)^{^{-24}}$ This one may be intimidating but $2=\frac{1}{2}^{-1}$ .

Therefore,

$\left(\left[\frac{1}{2}\right]^{-24}\right)^{-1}=2^{24}$

$2^{24}$

is not like the answers so this is the correct answer.

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Question

find x

8x=2x+6

Answer

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

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Question

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

Answer

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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Question

Solve for .

Answer

can be written as

Since there is a common base of , we can say

or .

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Question

Solve for .

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

Answer

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

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Question

Solve for .

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

Answer

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.

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

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Question

Solve for

$2^{x+1}=128$

Answer

Recall that .

With same base, we can write this equation:

.

By subtracting on both sides, .

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Question

Solve for .

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

Answer

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 .

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Question

Solve for .

Answer

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

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

$2\cdot 4=8$ .

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Question

Solve for .

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

Answer

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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Question

Solve for .

Answer

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 $x=\frac{1}{10}$ .

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Question

Solve for .

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

Answer

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

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 $x=\frac{-1}{10}$ .

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