Question 1

According to the product law of exponents, when multiplying powers with the same base, you _____ the exponents.

• Correct Answer: add
• Distractors: subtract, multiply, divide
• Correct Feedback: The product law states that $a^m \cdot a^n = a^{m+n}$, so the exponents are added.
• Incorrect Feedback: Exponents are added during multiplication of like bases, not subtracted, multiplied, or divided.

Question 2

According to the quotient law of exponents, when dividing powers with the same base, you _____ the exponents.

• Correct Answer: subtract
• Distractors: add, multiply, divide
• Correct Feedback: The quotient law states that $\frac{a^m}{a^n} = a^{m-n}$, so the bottom exponent is subtracted from the top exponent.
• Incorrect Feedback: Exponents are subtracted when dividing powers of the same base, not added, multiplied, or divided.

Question 3

Any non-zero real number raised to the power of zero is always equal to _____ .

• Correct Answer: 1
• Distractors: 0, -1, infinity
• Correct Feedback: By the zero exponent rule, $a^0 = 1$ for any non-zero real number $a$.
• Incorrect Feedback: Any non-zero base raised to power 0 evaluates to 1, not 0, -1, or infinity.

Question 4

The value of the expression $5^{-2}$ expressed as a positive fraction is $\frac{1}{\text{\_\_\_\_\_}}$.

• Correct Answer: 25
• Distractors: 10, -25, -10
• Correct Feedback: Using the negative exponent rule $a^{-n} = \frac{1}{a^n}$, $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$.
• Incorrect Feedback:$5^{-2}$ equals $\frac{1}{5^2} = \frac{1}{25}$, not 10 or a negative value.

Question 5

The fractional exponent expression $16^{\frac{1}{2}}$ evaluates to the integer _____ .

• Correct Answer: 4
• Distractors: 8, 2, 32
• Correct Feedback: An exponent of $\frac{1}{2}$ represents the square root, so $16^{\frac{1}{2}} = \sqrt{16} = 4$.
• Incorrect Feedback:$16^{\frac{1}{2}}$ is the square root of 16, which is 4, not 8, 2, or 32.

Question 6

When raising a power to another power, as in $(a^m)^n$, you _____ the exponents.

• Correct Answer: multiply
• Distractors: add, subtract, divide
• Correct Feedback: The power of a power rule states that $(a^m)^n = a^{m \cdot n}$, meaning you multiply the exponents.
• Incorrect Feedback: When raising a power to a power, exponents are multiplied together.

Question 7

Simplifying the expression $(27)^{\frac{2}{3}}$ gives the final numerical value _____ .

• Correct Answer: 9
• Distractors: 3, 18, 81
• Correct Feedback:$27^{\frac{2}{3}} = (27^{\frac{1}{3}})^2 = 3^2 = 9$.
• Incorrect Feedback: The cube root of 27 is 3, and squaring 3 gives 9.

Question 8

The expression $(\frac{8}{125})^{-\frac{1}{3}}$ simplifies to the improper fraction $\frac{5}{\text{\_\_\_\_\_}}$.

• Correct Answer: 2
• Distractors: 4, 8, 25
• Correct Feedback:$(\frac{8}{125})^{-\frac{1}{3}} = (\frac{125}{8})^{\frac{1}{3}} = \frac{5}{2}$. The denominator is 2.
• Incorrect Feedback: Inverting the base due to the negative power gives 125/8, and taking the cube root results in 5/2.

Question 9

When simplifying $x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}$, the single simplified exponent on $x$ is $\frac{1}{\text{\_\_\_\_\_}}$.

• Correct Answer: 2
• Distractors: 4, 3, 8
• Correct Feedback: Adding the exponents gives $\frac{3}{4} + (-\frac{1}{4}) = \frac{2}{4} = \frac{1}{2}$.
• Incorrect Feedback: Adding the fractional exponents yields 2/4, which simplifies to 1/2.

Question 10

The value of $32^{-0.4}$ expressed as a unit fraction is $\frac{1}{\text{\_\_\_\_\_}}$.

• Correct Answer: 4
• Distractors: 2, 8, 16
• Correct Feedback: Since $-0.4 = -\frac{2}{5}$, $32^{-\frac{2}{5}} = (32^{\frac{1}{5}})^{-2} = 2^{-2} = \frac{1}{4}$.
• Incorrect Feedback: Convert -0.4 to -2/5; 32 to the power of 1/5 is 2, and 2 to the power -2 is 1/4.

Question 11

Simplifying $\frac{y^2}{y^{-3}}$ yields $y$ raised to the positive exponent _____ .

• Correct Answer: 5
• Distractors: -1, 1, -5
• Correct Feedback: Using the quotient rule, subtract the exponent in the denominator: $2 – (-3) = 2 + 3 = 5$.
• Incorrect Feedback: Subtracting negative 3 from 2 results in 2 + 3 = 5.

Question 12

The numerical expression $(2^3 \cdot 3^3)^{\frac{1}{3}}$ evaluates to _____ .

• Correct Answer: 6
• Distractors: 12, 18, 24
• Correct Feedback: Distributing the power yields $2^{3 \cdot \frac{1}{3}} \cdot 3^{3 \cdot \frac{1}{3}} = 2^1 \cdot 3^1 = 6$.
• Incorrect Feedback: Applying the exponent 1/3 to each base gives 2 * 3 = 6.

Question 13

The expression $(\frac{64}{729})^{-\frac{2}{3}}$ simplifies to the ratio $\frac{81}{\text{\_\_\_\_\_}}$.

• Correct Answer: 16
• Distractors: 9, 36, 64
• Correct Feedback:$(\frac{64}{729})^{-\frac{2}{3}} = (\frac{729}{64})^{\frac{2}{3}} = (\frac{9}{4})^2 = \frac{81}{16}$.
• Incorrect Feedback: Inverting gives 729/64; taking the cube root yields 9/4, and squaring gives 81/16.

Question 14

If $2^{x-1} \cdot 4^{x+1} = 32$, the value of $x$ is $\frac{4}{\text{\_\_\_\_\_}}$.

• Correct Answer: 3
• Distractors: 2, 5, 4
• Correct Feedback: Convert to base 2: $2^{x-1} \cdot 2^{2(x+1)} = 2^5$, so $x-1+2x+2 = 5$ which gives $3x+1=5$ or $x = \frac{4}{3}$.
• Incorrect Feedback: Expressing all terms with base 2 gives 3x + 1 = 5, leading to x = 4/3.

Question 15

The simplified form of $\frac{9^{\frac{3}{2}} \cdot 27^{-\frac{2}{3}}}{3^2}$ is $\frac{1}{\text{\_\_\_\_\_}}$.

• Correct Answer: 3
• Distractors: 9, 27, 1
• Correct Feedback: Express in powers of 3: $9^{\frac{3}{2}} = 3^3 = 27$, $27^{-\frac{2}{3}} = 3^{-2} = \frac{1}{9}$. Numerator is $3^3 \cdot 3^{-2} = 3^1$. Thus $\frac{3^1}{3^2} = 3^{-1} = \frac{1}{3}$.
• Incorrect Feedback: Converting to base 3 yields 3^(3 – 2 – 2) = 3^-1 = 1/3.

Question 16

The value of the expression $(1^3 + 2^3 + 3^3)^{-\frac{1}{2}}$ is $\frac{1}{\text{\_\_\_\_\_}}$.

• Correct Answer: 6
• Distractors: 36, 12, 18
• Correct Feedback: Inside terms sum to $1 + 8 + 27 = 36$. Then $36^{-\frac{1}{2}} = \frac{1}{\sqrt{36}} = \frac{1}{6}$.
• Incorrect Feedback: Summing the cubes gives 36; taking the negative one-half power gives 1/sqrt(36) = 1/6.

Question 17

Simplifying $\frac{a^{-2} b^3}{a^3 b^{-1}}$ to have only positive exponents yields $\frac{b^4}{a^{\text{\_\_\_\_\_}}}$.

• Correct Answer: 5
• Distractors: 1, -5, 6
• Correct Feedback: For base $a$: $a^{-2-3} = a^{-5} = \frac{1}{a^5}$. For base $b$: $b^{3-(-1)} = b^4$. Result is $\frac{b^4}{a^5}$.
• Incorrect Feedback: Moving negative exponents gives b^(3+1) in numerator and a^(3+2) = a^5 in denominator.

Question 18

The product $\sqrt{x} \cdot \sqrt[3]{x} \cdot \sqrt[6]{x}$ simplifies to $x$ raised to the power _____ .

• Correct Answer: 1
• Distractors: 2, 1/2, 6
• Correct Feedback: Convert roots to fractional exponents: $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3+2+1}{6} = \frac{6}{6} = 1$.
• Incorrect Feedback: Adding the fractional exponents 1/2, 1/3, and 1/6 gives 6/6 = 1, so the result is x^1.