Exponent Expressions - SSAT Upper Level Quantitative

1. Which of the following is equivalent to $\left(\frac{2^3}{2^{-1}}\right)^2 \div 2^4$?
   Answer: $2^4$ 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 $2^8 \div 2^4$ as $2^{8-2}$. 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.
2. For which value of $k$ is $(2^k)^3 \cdot 2^{-5} = \frac{1}{4}$?
   Answer: $1$ 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 $(2^2)^3 \cdot 2^{-5} = 2^6 \cdot 2^{-5} = 2^1 = 2$.
3. If $a = 2^3$ and $b = 2^4$, what is the value of $\frac{a^2 \cdot b^3}{(ab)^2}$?
   Answer: 16 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 $2^4$ as $2^5$.
4. The expression $\frac{(3^2)^x \cdot 3^{-x}}{3^{x-1}}$ simplifies to:
   Answer: $3$ 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 $3^1$ as $3^2$. 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.

Start this quiz