SSAT Upper Level Quiz: Exponent Expressions

4 questions · exam conditions

0:00

Exponent ExpressionsQuestion 1 of 4

The expression $\frac{(3^2)^x \cdot 3^{-x}}{3^{x-1}}$ simplifies to:

3

9

\frac{1}{3}

3^x

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SSAT Upper Level Quiz

SSAT Upper Level Quiz: Exponent Expressions

Practice Exponent Expressions in SSAT Upper Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Exponent Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for SSAT Upper Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The expression $\frac{(3^2)^x \cdot 3^{-x}}{3^{x-1}}$ simplifies to:

$3$ (correct answer)
$9$
$\frac{1}{3}$
$3^x$

Explanation: First, simplify the numerator:

(3^2)^x \cdot 3^{-x} = 3^{2x} \cdot 3^{-x} = 3^{2x-x} = 3^x

. So the expression becomes

\frac{3^x}{3^{x-1}}

. Using the quotient rule:

3^{x-(x-1)} = 3^{x-x+1} = 3^1 = 3

. The expression simplifies to the constant 3, regardless of the value of

x

. Choice B (9) might result from incorrectly calculating

3^1

3^2

. Choice C (

\frac{1}{3}

) could come from sign errors in the exponent arithmetic. Choice D (

3^x

) would result from incorrectly simplifying the quotient rule, perhaps by subtracting exponents in the wrong direction or forgetting to apply the rule entirely.

Question 2

If $a = 2^3$ and $b = 2^4$ , what is the value of $\frac{a^2 \cdot b^3}{(ab)^2}$ ?

4
8
16 (correct answer)
32

Explanation: Given

a = 2^3 = 8

and

b = 2^4 = 16

. We need to find

\frac{a^2 \cdot b^3}{(ab)^2}

. Using exponent rules:

\frac{a^2 \cdot b^3}{(ab)^2} = \frac{a^2 \cdot b^3}{a^2 \cdot b^2} = \frac{b^3}{b^2} = b^1 = b = 2^4 = 16

. Alternatively, substituting the values directly:

a^2 = (2^3)^2 = 2^6 = 64

b^3 = (2^4)^3 = 2^{12} = 4096

, and

(ab)^2 = (2^3 \cdot 2^4)^2 = (2^7)^2 = 2^{14}

. So

\frac{a^2 \cdot b^3}{(ab)^2} = \frac{2^6 \cdot 2^{12}}{2^{14}} = \frac{2^{18}}{2^{14}} = 2^4 = 16

. Choice A (4) would result from incorrectly calculating

b^3/b^2 = b

but using

a

instead of

b

. Choice B (8) might come from errors in exponent arithmetic. Choice D (32) could result from miscalculating

2^4

2^5

Question 3

Which of the following is equivalent to $\left(\frac{2^3}{2^{-1}}\right)^2 \div 2^4$ ?

$2^4$ (correct answer)
$2^6$
$2^8$
$2^{12}$

Explanation: First, simplify the expression inside the parentheses:

\frac{2^3}{2^{-1}} = 2^3 \cdot 2^1 = 2^{3-(-1)} = 2^{3+1} = 2^4

. Then:

\left(\frac{2^3}{2^{-1}}\right)^2 = (2^4)^2 = 2^8

. Finally:

2^8 \div 2^4 = 2^{8-4} = 2^4

. Choice B (

2^6

) might result from incorrectly calculating

2^8 \div 2^4

2^{8-2}

. Choice C (

2^8

) would result from forgetting to divide by

2^4

at the end. Choice D (

2^{12}

) could come from incorrectly multiplying exponents instead of using the quotient rule in the final step.

Question 4

For which value of $k$ is $(2^k)^3 \cdot 2^{-5} = \frac{1}{4}$ ?

$-1$
$0$
$1$ (correct answer)
$2$

Explanation: First, simplify the left side:

(2^k)^3 \cdot 2^{-5} = 2^{3k} \cdot 2^{-5} = 2^{3k-5}

. Since

\frac{1}{4} = \frac{1}{2^2} = 2^{-2}

, we have the equation

2^{3k-5} = 2^{-2}

. Therefore

3k - 5 = -2

, which gives us

3k = 3

, so

k = 1

. Let's verify: when

k = 1

(2^1)^3 \cdot 2^{-5} = 2^3 \cdot 2^{-5} = 2^{-2} = \frac{1}{4}

✓. Choice A would give

(2^{-1})^3 \cdot 2^{-5} = 2^{-3} \cdot 2^{-5} = 2^{-8} = \frac{1}{256}

. Choice B would give

(2^0)^3 \cdot 2^{-5} = 1 \cdot 2^{-5} = \frac{1}{32}

. Choice D would give

(2^2)^3 \cdot 2^{-5} = 2^6 \cdot 2^{-5} = 2^1 = 2