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Integer and rational exponents — ACT practice questions

Mathematics
45 questions on the paper
41 practice questions in the app
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Practice 41 Integer and rational exponents questions in the app

What this topic is

Integer and rational exponents on the ACT Mathematics test cover integer, negative, and fractional powers and scientific notation.

Students must rewrite products of powers, compare exponential quantities, and simplify expressions that hold for all nonzero u and v. Items often ask which listed form is equivalent, which quantity is greatest, or how many liters remain after 4.8 × 10^6 liters drain at 7.5 × 10^4 liters per hour for 8 hours.

Answer choices are short numeric or algebraic expressions, and traps include adding exponents that should be multiplied, mishandling negative exponents, and writing scientific notation with the wrong power of ten.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Easier

Which of the following is equivalent to $\dfrac{18u^{12}v^5}{3u^4v^2}$ for all nonzero $u$ and $v$ ?

6u^3v^3

✗ Not this one

15u^8v^3

✗ Not this one

6u^{16}v^7

✗ Not this one

6u^8v^3

✓ Correct

Show answer

Answer: D — $6u^8v^3$

The coefficient is $18\div3=6$ . Subtracting exponents gives $u^{12-4}v^{5-2}=u^8v^3$ , so the ratio is $6u^8v^3$ .

Question 2Easier

Which of the following is the greatest?

9^{-1/2}

✗ Not this one

8^{-1/3}

✓ Correct

16^{-1}

✗ Not this one

25^{-1/2}

✗ Not this one

Show answer

Answer: B — $8^{-1/3}$

The four values are $\dfrac{1}{3}$ , $\dfrac{1}{2}$ , $\dfrac{1}{16}$ , and $\dfrac{1}{5}$ , respectively. The greatest is $8^{-1/3}=\dfrac{1}{2}$ .

Question 3Easier

A reservoir initially contains $4.8 \times 10^6$ liters of water. Water drains at a constant rate of $7.5 \times 10^4$ liters per hour for 8 hours. How many liters remain, expressed in scientific notation?

4.725 \times 10^6

✗ Not this one

5.4 \times 10^6

✗ Not this one

4.2 \times 10^6

✓ Correct

4.2 \times 10^5

✗ Not this one

Show answer

Answer: C — $4.2 \times 10^6$

The amount drained is $(7.5\times10^4)(8)=6\times10^5$ liters. Rewriting this as $0.6\times10^6$ gives $(4.8-0.6)\times10^6=4.2\times10^6$ liters remaining.

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 41 Integer and rational exponents questions in the app

Integer and rational exponents — ACT practice questions

What this topic is

Sample questions

More ACT topics

Mathematics