Exponent Rules, Radicals and Rational Exponents: Algebra 1 Math Worksheet – PDF Download

Problem 1: Evaluate powers

Evaluate each expression.

1. \(3^{4}\)
2. \((-2)^{3}\)
3. \(-5^{2}\)

Problem 2: Pattern of powers of 10

Complete the table of powers of 10. Each step to the right divides by 10.

Problem 3: Product rule

Simplify each expression.

1. \(x^{6}\cdot x^{3}\)
2. \(a\cdot a^{4}\)
3. \(y^{2}\cdot y^{5}\cdot y\)

Problem 4: Quotient rule

Simplify each expression. Assume no variable equals 0.

1. \(\dfrac{x^{12}}{x^{5}}\)
2. \(\dfrac{m^{7}}{m}\)
3. \(\dfrac{b^{6}}{b^{6}}\)

Problem 5: Power of a power

Simplify each expression.

1. \((y^{4})^{3}\)
2. \((n^{5})^{2}\)
3. \((z^{2})^{6}\)

Problem 6: Power of a product

Simplify each expression.

1. \((2a^{3})^{4}\)
2. \((-3x^{2})^{3}\)
3. \((5pq^{2})^{2}\)

Problem 7: Coefficients and variables

Simplify each product.

1. \(3x^{2}\cdot4x^{5}\)
2. \((-2a^{3}b)(5ab^{4})\)
3. \((7y^{2})(-y^{6})\)

Problem 8: Dividing monomials

Simplify each quotient. Assume no variable equals 0.

1. \(\dfrac{20m^{8}}{5m^{3}}\)
2. \(\dfrac{36x^{5}y^{4}}{12xy^{2}}\)
3. \(\dfrac{-18a^{9}}{6a^{9}}\)

Problem 9: Negative exponents

Rewrite each expression with positive exponents and simplify.

1. \(4x^{-3}\)
2. \(\dfrac{5}{m^{-2}}\)
3. \(2^{-5}\)

Problem 10: Mixed exponent rules

Simplify. Write each answer with positive exponents.

1. \(x^{5}\cdot x^{-8}\)
2. \((2x^{3})^{2}\cdot x^{-4}\)
3. \(\dfrac{a^{-2}b^{5}}{a^{3}b^{2}}\)

Problem 11: Unit-fraction exponents

Evaluate each expression.

1. \(25^{1/2}\)
2. \(64^{1/3}\)
3. \(81^{1/4}\)

Problem 12: Radicals to exponents

Write each expression using a rational exponent.

1. \(\sqrt{7}\)
2. \(\sqrt[3]{x^{5}}\)
3. \((\sqrt[4]{a})^{3}\)

Problem 13: Exponents to radicals

Write each expression as a radical.

1. \(x^{1/2}\)
2. \(y^{2/3}\)
3. \(5^{3/4}\)

Problem 14: Rational powers

Evaluate each expression. Use the figure for part (b).

1. \(8^{2/3}\)
2. \(4^{3/2}\)
3. \(81^{3/4}\)

Problem 15: Negative rational powers

Evaluate each expression.

1. \(27^{-2/3}\)
2. \(16^{-1/2}\)
3. \(32^{-3/5}\)

Problem 16: Reading a graph

The graph shows \(y=x^{1/2}\).

1. Find \(9^{1/2}\) from the graph.
2. Use the graph to estimate \(10^{1/2}\).
3. Find the value of \(x\) when \(y=4\).

Problem 17: Simplify numeric square roots

Write each radical in simplest form.

1. \(\sqrt{50}\)
2. \(\sqrt{48}\)
3. \(\sqrt{200}\)

Problem 18: Rational exponent rules

Simplify. Assume \(x>0\).

1. \(x^{1/2}\cdot x^{3/2}\)
2. \(\dfrac{x^{5/3}}{x^{2/3}}\)
3. \((x^{3/4})^{2}\)
4. \((x^{2/3})^{6}\)

Problem 19: Square roots with variables

Simplify. Assume every variable is positive.

1. \(\sqrt{x^{6}}\)
2. \(\sqrt{18x^{4}}\)
3. \(\sqrt{75a^{5}}\)

Problem 20: Square garden

Maya’s square garden has an area of 98 square feet. Find the side length in simplest radical form and to the nearest tenth of a foot. Then find the length of fencing needed to go around the garden, to the nearest tenth.

Problem 21: Orbit model

In a model of a distant star system, a planet that orbits at distance \(d\) astronomical units (AU) from its star has a period of \(T=d^{3/2}\) years.

1. Find \(T\) when \(d=4\) and when \(d=9\). Compare with the graph.
2. A second planet has a period of 64 years. Find its distance \(d\).

Problem 22: Cube crate

A cube-shaped crate holds 512 cubic inches. Find the edge length using a rational exponent. Then find the total outside surface area of the crate.

Problem 23: Patio diagonal

Ethan lays a square patio with 6-foot sides. He wants a rope along the diagonal. Use the Pythagorean theorem to find the diagonal length in simplest radical form and to the nearest tenth.

Problem 24: Find the errors

Two students made mistakes. Explain each mistake and give the correct answer.

1. Jamal wrote \((2x^{3})^{3}=6x^{9}\).
2. Aisha wrote \(16^{3/2}=24\) because \(16\cdot\frac{3}{2}=24\).

Problem 25: Cube with a variable edge

A cube has volume \(27x^{6}\) cubic units, where \(x>0\). Find the edge length as a simplified expression. Then write the surface area of the cube.

See every chapter on the Algebra 1 math page.