Which of the following is equal to (2^3)^2 × 2^-4?

B. 4. Raise a power to a power by multiplying the exponents: (2^3)^2 = 2^6. Multiply powers with the same base by adding the exponents: 2^6 × 2^-4 = 2^2. Evaluate: 2^2 = 4.

Vectors u = and v = are given in component form. What is the magnitude of 2u + v?

A. 5. Scale first: 2u = . Add the components: + = . Find the magnitude: sqrt(4^2 + 3^2) = sqrt(25) = 5.

Which of the following is equal to (5 + 5i)/(1 + 2i), where i = sqrt(-1)?

D. 3 - i. Multiply the numerator and the denominator by the conjugate of the denominator, 1 - 2i. Numerator: (5 + 5i)(1 - 2i) = 5 - 10i + 5i - 10i^2 = 15 - 5i, because i^2 = -1. Denominator: (1 + 2i)(1 - 2i) = 1 - 4i^2 = 5. Divide: (15 - 5i)/(5) = 3 - i.

ACT Math reporting category

ACT Number & Quantity Practice

Number & Quantity is the ACT math reporting category about how numbers behave: real and complex numbers, exponents, radicals, and intro vectors and matrices. According to ACT, it makes up 10–12% of scored math questions. It is one of five areas under Preparing for Higher Math.

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10–12% of scored Math questions3 original questionsInstant answer checks

At a glance

Number & Quantity on the ACT: the key numbers

Section and category facts from ACT.org, checked October 9, 2026. Time per question is ACT's section time divided by its question count.
Section	ACT Math: 45 questions in 50 minutes
Scored questions	41 scored, 4 field test
Reporting category	Number & Quantity (part of Preparing for Higher Math)
Share of scored questions	10–12%, according to ACT
Average time per question	About 67 seconds
Counts toward the Composite	Yes
Practice on this page	3 original questions: Easy, Medium, Hard

What this tests

What does ACT Number & Quantity test?

Exponents and radicals. Simplify powers, negative and fractional exponents, and square roots, and rewrite expressions between radical form and exponent form.
The real number system. Compare and order integers, fractions, decimals, and irrational numbers, and tell rational values from irrational ones.
Complex numbers. Add, subtract, multiply, and divide numbers of the form $a + bi$ , using $i^{2} = - 1$ and conjugates.
Vectors and matrices. Work with vectors in component form and with basic matrix operations, such as adding matrices or multiplying a matrix by a number.
Number properties. Use factors, multiples, primes, divisibility, and odd and even rules to decide what must be true about a number.

Practice questions

Practice ACT Number & Quantity questions

Pick an answer to check it. Each question has one correct choice.

Question 1 of 3Easy

Which of the following is equal to

(2^{3})^{2} \cdot 2^{- 4}

Show answer and explanation

Correct answerB. $4$

How to solve it

Raise a power to a power by multiplying the exponents: $(2^{3})^{2} = 2^{6}$ .
Multiply powers with the same base by adding the exponents: $2^{6} \cdot 2^{- 4} = 2^{2}$ .
Evaluate: $2^{2} = 4$ .

Why the other choices miss

2

comes from adding the exponents 3 and 2 instead of multiplying them.

32

comes from squaring the exponent 3 to get

2^{9}

1024

comes from dropping the negative sign and adding 6 and 4.

Question 2 of 3Medium

Vectors

u = ⟨ 3, - 1 ⟩

and

v = ⟨ - 2, 5 ⟩

are given in component form. What is the magnitude of

2 u + v

Show answer and explanation

Correct answerA. $5$

How to solve it

Scale first: $2 u = ⟨ 6, - 2 ⟩$ .
Add the components: $⟨ 6, - 2 ⟩ + ⟨ - 2, 5 ⟩ = ⟨ 4, 3 ⟩$ .
Find the magnitude: $4^{2} + 3^{2} = 25 = 5$ .

Why the other choices miss

7

adds the components 4 and 3 instead of using the Pythagorean theorem.

217

doubles both vectors and finds the length of

⟨ 2, 8 ⟩

210 + 29

adds the lengths of

2 u

and

v

, which works only when vectors point the same way.

Question 3 of 3Hard

Which of the following is equal to

\frac{5 + 5 i}{1 + 2 i}

, where

i = - 1

Show answer and explanation

Correct answerD. $3 - i$

How to solve it

Multiply the numerator and the denominator by the conjugate of the denominator, $1 - 2 i$ .
Numerator: $(5 + 5 i) (1 - 2 i) = 5 - 10 i + 5 i - 10 i^{2} = 15 - 5 i$ , because $i^{2} = - 1$ .
Denominator: $(1 + 2 i) (1 - 2 i) = 1 - 4 i^{2} = 5$ .
Divide: $\frac{15 - 5 i}{5} = 3 - i$ .

Why the other choices miss

- 1 - i

treats

- 10 i^{2}

- 10

instead of

+ 10

- 1 + 3 i

multiplies the numerator by

1 + 2 i

instead of the conjugate

1 - 2 i

5 + \frac{5}{2} i

divides the real parts and the imaginary parts separately, which is not how complex division works.

Avoid these traps

What mistakes do students make on Number & Quantity?

Adding exponents in powers

For a power raised to a power, like $(x^{3})^{2}$ , students add the exponents and get $x^{5}$ instead of multiplying them to get $x^{6}$ .

Fix: Add exponents when multiplying powers with the same base. Multiply exponents when raising a power to a power.

Treating i squared as positive

When expanding products of complex numbers, students write $i^{2}$ as $1$ , which flips the sign of part of the real term.

Fix: Replace every $i^{2}$ with $- 1$ right after expanding, then combine the real parts and the imaginary parts separately.

Adding vector lengths

Students find the length of a vector sum by adding the two lengths. That works only when both vectors point in the same direction.

Fix: Add the vectors component by component first, then apply $x^{2} + y^{2}$ to the result.

Study method

How should you practice ACT Number & Quantity?

Write the exponent rules from memory (product, quotient, power of a power, negative and fractional exponents), then check them against your notes.
Work exponent and radical problems by hand, one rule per line, before trying shortcuts.
For complex numbers, expand fully, replace $i^{2}$ with $- 1$ , and multiply by the conjugate to divide.
Keep an error log that names the rule you broke on each miss, not just the correct answer.
Mix Number & Quantity questions with other math topics in Dolphin so you practice spotting which rule a problem needs.

Start ACT practice in Dolphin

FAQ

Questions about ACT Number & Quantity

What does Number & Quantity cover on the ACT?

It covers how numbers work: real and complex numbers, integer and fractional exponents, radicals, intro vectors and matrices, and number properties such as factors and divisibility. The practice questions on this page cover three of those skills: exponent rules, vector magnitude, and dividing complex numbers.

How does Number & Quantity fit into Preparing for Higher Math?

ACT groups five reporting categories under Preparing for Higher Math: Number & Quantity, Algebra, Functions, Geometry, and Statistics & Probability. According to ACT, the group makes up 80% of scored math questions. Number & Quantity has the smallest listed share of the five, at 10–12%.

How many questions are on the ACT math section?

According to ACT, the math section has 45 questions in 50 minutes: 41 scored questions and 4 field-test questions. That works out to about 67 seconds per question. At 10–12% of the 41 scored questions, Number & Quantity comes to roughly 4 or 5 questions.

Will my score report show how I did on Number & Quantity?

Yes. ACT's score report lists each reporting category with the number of questions in it, the number you answered correctly, and your percent correct. Use that line to decide whether exponents, complex numbers, and related skills need more practice.

Should I guess on a Number & Quantity question I cannot solve?

Yes. ACT does not penalize incorrect answers, so leaving a question blank can only cost you. Eliminate any choice that breaks a rule you know, such as a sign error with $i^{2}$ , then pick from what is left and move on.

Related practice

Sources

Where these facts come from

ACT: Understanding Your ACT Scores: reporting categories, category shares, Composite calculation, and scoring.
ACT: Test Enhancements and Changes: section length, question counts, and optional Science.
ACT: Official test preparation and free practice tests: released ACT questions in the real test format.

ACT facts checked October 9, 2026. Every practice question on this page is original Dolphin practice written to match one ACT reporting category. None is a released ACT item.

ACT® is a trademark registered by ACT, Inc., which is not affiliated with, and does not endorse, Dolphin.

ACT Number & Quantity Practice

Number & Quantity on the ACT: the key numbers

What does ACT Number & Quantity test?

Practice ACT Number & Quantity questions

How to solve it

How to solve it

How to solve it

What mistakes do students make on Number & Quantity?

Adding exponents in powers

Treating i squared as positive

Adding vector lengths

How should you practice ACT Number & Quantity?

Questions about ACT Number & Quantity

Keep practicing for the ACT

Where these facts come from