Algebra 1 Exponent Rules and Radicals Worksheet (PDF)

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

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This exponent rules worksheet 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.

Use these 20 problems to master exponent rules and radicals (Common Core N-RN.A.1, N-RN.A.2, and N-RN.B.3). You will use product, quotient, and power rules, work with zero and negative exponents, simplify radicals, and classify rational and irrational numbers. They move from quick warm-ups to multistep word problems, so slow down on the last six and check that each answer makes sense.

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

Exponent rules worksheet: exercises 1 to 10

Exercise 1 – Evaluate a power

Four factors of 3.

Evaluate \(3^{4}\).

Exercise 2 – Product rule

Simplify \(x^{5} \cdot x^{3}\) and evaluate \(2^{3} \cdot 2^{4}\).

Exercise 3 – Quotient rule

Simplify \(\frac{x^{9}}{x^{4}}\).

Exercise 4 – Power of a power

Simplify \((x^{3})^{4}\).

Exercise 5 – Zero exponent

Evaluate \(7^{0}\), \((-5)^{0}\), and \((5x)^{0}\) for \(x \ne 0\).

Exercise 6 – Negative exponent

Each step to the right divides by 3.

Evaluate \(3^{-2}\).

Exercise 7 – Power of a product

Simplify \((2x^{4})^{3}\).

Exercise 8 – Negative exponent in a term

Write \(4x^{-2}\) with a positive exponent.

Exercise 9 – Quotient of monomials

Simplify \(\frac{6x^{5}y^{2}}{2x^{2}y}\).

Exercise 10 – Several rules

Simplify \((2x^{3}y^{-2})^{2}\) using positive exponents.

Exercises 11 and up

Exercise 11 – Simplify a radical

Simplify \(\sqrt{48}\).

Exercise 12 – Simplify another radical

Choose the largest perfect square.

Rewrite \(\sqrt{200}\) in simplest radical form.

Exercise 13 – Rational exponents

Evaluate \(25^{1/2}\), \(27^{2/3}\), and \(16^{3/4}\).

Exercise 14 – Radical and exponent forms

Write \(5^{1/2}\) as a radical and \(\sqrt[3]{x^{2}}\) with a rational exponent.

Exercise 15 – Rational or irrational

Numbers on the number line.

Classify each number as rational or irrational: \(\sqrt{9}\), \(\sqrt{10}\), \(0.\overline{3}\), \(\pi\).

Exercise 16 – Bacteria

The first hours.

A culture starts with 100 bacteria and doubles every hour, so \(P = 100 \cdot 2^{t}\) after \(t\) hours. Find the population after 6 hours.

Exercise 17 – Computer memory

One kilobyte is \(2^{10}\) bytes, and one megabyte is \(2^{10}\) kilobytes. Use the product rule to write the number of bytes in one megabyte as a power of 2, then evaluate.

Exercise 18 – Square garden

A square garden has area 75 square feet. Find the side length in simplest radical form and as a decimal to the nearest hundredth.

Exercise 19 – Falling object

The time in seconds for an object to fall \(h\) feet is \(t = \frac{\sqrt{h}}{4}\). Find the time to fall 64 feet and 50 feet (simplest radical form).

Exercise 20 – Explain irrational sums

Explain why \(\frac{1}{2} + \sqrt{2}\) is irrational.

Exercise 21 – Sunlight travel time

Light travels about \(1.86 \times 10^{5}\) miles per second. The Sun is about \(9.3 \times 10^{7}\) miles away. Use the quotient rule to find the travel time in seconds.

Key facts behind this exponent rules worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

All Algebra 1 chapters: Algebra 1 math.

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