Algebra 1 Exponent Rules and Radicals Worksheet Solutions (PDF)

Algebra 1 math worksheet solutions: Exponent Rules and Radicals – Maths PDF

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This exponent rules answers for Algebra 1 covers exponent rules and radicals (Common Core N-RN.A.1, N-RN.A.2, N-RN.B.3), with a free PDF to print.

This answer key explains every problem on the exponent rules worksheet. Final answers are in bold, and the lines above them show how each result was found. Use it to correct mistakes and to look for patterns in the errors you make most often.

Algebra 1 · Common Core State Standards: N-RN.A.1, N-RN.A.2, N-RN.B.3

Exponent rules answers: exercises 1 to 10

Exercise 1 – Evaluate a power

\(3^{4} = 3 \cdot 3 \cdot 3 \cdot 3 = 81\).

Exercise 2 – Product rule

Add the exponents: \(x^{5 + 3} = x^{8}\). And \(2^{3} \cdot 2^{4} = 2^{7} = 128\).

\(x^{8}\); 128

Exercise 3 – Quotient rule

Subtract the exponents: \(x^{9 – 4} = x^{5}\).

\(x^{5}\)

Exercise 4 – Power of a power

Multiply the exponents: \(x^{3 \cdot 4} = x^{12}\).

\(x^{12}\)

Exercise 5 – Zero exponent

Any nonzero base raised to the power 0 equals 1.

1, 1, and 1

Exercise 6 – Negative exponent

\(3^{-2} = \frac{1}{3^{2}} = \frac{1}{9}\).

\(\frac{1}{9}\)

Exercise 7 – Power of a product

Apply the power to each factor: \(2^{3} \cdot (x^{4})^{3} = 8x^{12}\).

\(8x^{12}\)

Exercise 8 – Negative exponent in a term

Only \(x\) has the negative exponent: \(4x^{-2} = \frac{4}{x^{2}}\).

\(\frac{4}{x^{2}}\)

Exercise 9 – Quotient of monomials

Divide coefficients: \(6 \div 2 = 3\). Subtract exponents: \(x^{5-2} = x^{3}\), \(y^{2-1} = y\).

\(3x^{3}y\)

Exercise 10 – Several rules

Square each factor: \(2^{2}x^{6}y^{-4} = 4x^{6}y^{-4}\).

Move \(y^{-4}\) to the denominator.

\(\frac{4x^{6}}{y^{4}}\)

Exercises 11 and up

Exercise 11 – Simplify a radical

Pull out the perfect square.

The greatest perfect square factor is 16: \(\sqrt{48} = \sqrt{16 \cdot 3}\).

\(= 4\sqrt{3}\).

\(4\sqrt{3}\)

Exercise 12 – Simplify another radical

The greatest perfect square factor is 100: \(\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}\).

\(10\sqrt{2}\)

Exercise 13 – Rational exponents

\(25^{1/2} = \sqrt{25} = 5\).

\(27^{2/3} = (\sqrt[3]{27})^{2} = 3^{2} = 9\).

\(16^{3/4} = (\sqrt[4]{16})^{3} = 2^{3} = 8\).

5, 9, and 8

Exercise 14 – Radical and exponent forms

The exponent \(\frac{1}{2}\) means a square root: \(5^{1/2} = \sqrt{5}\).

The cube root of \(x^{2}\) is \(x^{2/3}\).

\(\sqrt{5}\); \(x^{2/3}\)

Exercise 15 – Rational or irrational

\(\sqrt{9} = 3\) is rational. \(\sqrt{10}\) is not a perfect square, so it is irrational. \(0.\overline{3} = \frac{1}{3}\) is rational. \(\pi\) is irrational.

rational, irrational, rational, irrational

Exercise 16 – Bacteria

\(P = 100 \cdot 2^{6} = 100 \cdot 64 = 6400\).

6,400 bacteria

Exercise 17 – Computer memory

\(2^{10} \cdot 2^{10} = 2^{20}\).

\(2^{20} = 1{,}048{,}576\).

\(2^{20}\) = 1,048,576 bytes

Exercise 18 – Square garden

A square of area 75 square feet.

Side = \(\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}\).

Since \(\sqrt{3} \approx 1.732\), the side is about \(5(1.732) \approx 8.66\).

\(5\sqrt{3}\) ft ≈ 8.66 ft

Exercise 19 – Falling object

\(t = \frac{\sqrt{64}}{4} = \frac{8}{4} = 2\) seconds.

\(t = \frac{\sqrt{50}}{4} = \frac{5\sqrt{2}}{4}\) seconds, about 1.77 seconds.

2 s; \(\frac{5\sqrt{2}}{4}\) s

Exercise 20 – Explain irrational sums

Suppose the sum were a rational number \(r\). Then \(\sqrt{2} = r – \frac{1}{2}\).

A difference of two rational numbers is rational, so \(\sqrt{2}\) would be rational. But \(\sqrt{2}\) is irrational. This contradiction shows the sum is irrational.

The sum is irrational

Exercise 21 – Sunlight travel time

Divide the numbers and the powers of ten separately.

\(\frac{9.3 \times 10^{7}}{1.86 \times 10^{5}} = \frac{9.3}{1.86} \times 10^{7-5}\).

\(= 5 \times 10^{2} = 500\).

500 seconds

Key facts behind this exponent rules answers

Key facts

Product rule: aᵐ · aⁿ = aᵐ⁺ⁿ; quotient rule: aᵐ / aⁿ = aᵐ⁻ⁿ.
Power rule: (aᵐ)ⁿ = aᵐⁿ.
a⁰ = 1 and a⁻ⁿ = 1/aⁿ for a ≠ 0.
a^(1/2) = √a, and a^(m/n) is the n-th root of a raised to the m.
A rational plus an irrational number is irrational.

How to use this exponent rules answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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