Properties of Exponents

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SSAT Upper Level Quantitative › Properties of Exponents

Questions 1 - 10

1

and are both positive integers; A is odd. What can you say about the number

$(A + B + 1)^{A+B+1}$ ?

$(A + B + 1)^{A+B+1}$ is odd if is odd, and even if is even.

$(A + B + 1)^{A+B+1}$ is even if is odd, and odd if is even.

$(A + B + 1)^{A+B+1}$ is odd if is odd, and can be odd or even if is even.

$(A + B + 1)^{A+B+1}$ is even if is odd, and can be odd or even if is even.

$(A + B + 1)^{A+B+1}$ is even if is even, and can be odd or even if is odd.

Explanation

If is odd, then , the sum of three odd integers, is odd; an odd number taken to any positive integer power is odd.

If is even, then , the sum of two odd integers and an even integer, is even; an even number taken to any positive integer power is even.

Therefore, $(A + B + 1)^{A+B+1}$ always assumes the same odd/even parity as .

2

and are both positive integers; A is odd. What can you say about the number

$(A + B + 1)^{A+B+1}$ ?

$(A + B + 1)^{A+B+1}$ is odd if is odd, and even if is even.

$(A + B + 1)^{A+B+1}$ is even if is odd, and odd if is even.

$(A + B + 1)^{A+B+1}$ is odd if is odd, and can be odd or even if is even.

$(A + B + 1)^{A+B+1}$ is even if is odd, and can be odd or even if is even.

$(A + B + 1)^{A+B+1}$ is even if is even, and can be odd or even if is odd.

Explanation

If is odd, then , the sum of three odd integers, is odd; an odd number taken to any positive integer power is odd.

If is even, then , the sum of two odd integers and an even integer, is even; an even number taken to any positive integer power is even.

Therefore, $(A + B + 1)^{A+B+1}$ always assumes the same odd/even parity as .

3

Evaluate: $\dpi{100} (3^{3})^{2}$

$\dpi{100} 3^{6}$

$\dpi{100} 3^{5}$

$\dpi{100} 243$

$\dpi{100} 729$

Explanation

A power raised to a power indicates that you multiply the two powers.

$\dpi{100} (3^{3})^{2}=3^{3\cdot 2}=3^{6}$

4

Evaluate: $\dpi{100} (3^{3})^{2}$

$\dpi{100} 3^{6}$

$\dpi{100} 3^{5}$

$\dpi{100} 243$

$\dpi{100} 729$

Explanation

A power raised to a power indicates that you multiply the two powers.

$\dpi{100} (3^{3})^{2}=3^{3\cdot 2}=3^{6}$

5

$\dpi{100} (0.75)^{2}=$

$\dpi{100} \frac{9}{16}$

$\dpi{100} 49.5$

$\dpi{100} 58.5$

$\dpi{100} 5.85$

Explanation

Convert .75 to a fraction. $\dpi{100} \frac{75}{100}=\frac{3}{4}$ .

Now multiply $\dpi{100} \frac{3}{4}\times \frac{3}{4}=\frac{3\times 3}{4\times 4}=\frac{9}{16}$

6

$\dpi{100} (0.75)^{2}=$

$\dpi{100} \frac{9}{16}$

$\dpi{100} 49.5$

$\dpi{100} 58.5$

$\dpi{100} 5.85$

Explanation

Convert .75 to a fraction. $\dpi{100} \frac{75}{100}=\frac{3}{4}$ .

Now multiply $\dpi{100} \frac{3}{4}\times \frac{3}{4}=\frac{3\times 3}{4\times 4}=\frac{9}{16}$

7

$\dpi{100} Evaluate: (0.50^{2})$

$\dpi{100} 0.25$

$\dpi{100} 1$

$\dpi{100} 25$

$\dpi{100} 2.5$

Explanation

We can either write $\dpi{100} 0.5\times 0.5$ , or we can convert this to a fraction and write

$\dpi{100} \frac{1}{2}\times \frac{1}{2}$

$\dpi{100} \frac{1}{2}\times \frac{1}{2}=\frac{1\times 1}{2\times 2}=\frac{1}{4}$

$\dpi{100} \frac{1}{4}$ in decimal form is 0.25.

8

$\dpi{100} Evaluate: (0.50^{2})$

$\dpi{100} 0.25$

$\dpi{100} 1$

$\dpi{100} 25$

$\dpi{100} 2.5$

Explanation

We can either write $\dpi{100} 0.5\times 0.5$ , or we can convert this to a fraction and write

$\dpi{100} \frac{1}{2}\times \frac{1}{2}$

$\dpi{100} \frac{1}{2}\times \frac{1}{2}=\frac{1\times 1}{2\times 2}=\frac{1}{4}$

$\dpi{100} \frac{1}{4}$ in decimal form is 0.25.

9

$(-4)^{2} + (-4)^{3}$

Explanation

$(-4)^{2} = -4 \cdot (-4) = +(4\cdot 4) = 16$

$(-4)^{3} = -4 \cdot (-4)\cdot (-4) = 16 \cdot (-4) = - (16 \cdot 4) = -64$

Therefore,

$(-4)^{2} + (-4)^{3} = 16 + (-64) = - (64-16) = -48$

10

$(-4)^{2} + (-4)^{3}$

Explanation

$(-4)^{2} = -4 \cdot (-4) = +(4\cdot 4) = 16$

$(-4)^{3} = -4 \cdot (-4)\cdot (-4) = 16 \cdot (-4) = - (16 \cdot 4) = -64$

Therefore,

$(-4)^{2} + (-4)^{3} = 16 + (-64) = - (64-16) = -48$

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