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

Problem 1: Evaluate powers

1. \(3^{4}=3\cdot3\cdot3\cdot3=\)\(81\)
2. \((-2)^{3}=(-2)(-2)(-2)=\)\(-8\)
3. The exponent applies only to 5, so \(-5^{2}=-(5\cdot 5)=\)\(-25\).

Problem 2: Pattern of powers of 10

Divide by 10 at each step: \(10^{0}=10: 10=\)\(1\), \(10^{-1}=1: 10=\)\(\frac{1}{10}\), \(10^{-2}=\)\(\frac{1}{100}\).

Problem 3: Product rule

1. Add the exponents: \(x^{6+3}=\)\(x^{9}\).
2. \(a^{1+4}=\)\(a^{5}\).
3. \(y^{2+5+1}=\)\(y^{8}\).

Problem 4: Quotient rule

1. Subtract: \(x^{12-5}=\)\(x^{7}\).
2. \(m^{7-1}=\)\(m^{6}\).
3. \(b^{6-6}=b^{0}=\)\(1\).

Problem 5: Power of a power

1. Multiply: \(y^{4\cdot3}=\)\(y^{12}\).
2. \(n^{10}\), so \(n^{10}\).
3. \(z^{2\cdot6}=\)\(z^{12}\).

Problem 6: Power of a product

1. \(2^{4}a^{12}=\)\(16a^{12}\).
2. \((-3)^{3}x^{6}=\)\(-27x^{6}\).
3. \(5^{2}p^{2}q^{4}=\)\(25p^{2}q^{4}\).

Problem 7: Coefficients and variables

1. Multiply coefficients, add exponents: \(12x^{7}\), so \(12x^{7}\).
2. \((-2)(5)=-10\), \(a^{3+1}=a^{4}\), \(b^{1+4}=b^{5}\). Answer: \(-10a^{4}b^{5}\).
3. \((7)(-1)=-7\) and \(y^{2+6}=y^{8}\). Answer: \(-7y^{8}\).

Problem 8: Dividing monomials

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

Problem 9: Negative exponents

1. Only \(x\) has the negative exponent: \(\dfrac{4}{x^{3}}\).
2. Move \(m^{-2}\) to the numerator: \(5m^{2}\).
3. \(2^{-5}=\dfrac{1}{2^{5}}=\)\(\dfrac{1}{32}\).

Problem 10: Mixed exponent rules

1. \(x^{5-8}=x^{-3}=\)\(\dfrac{1}{x^{3}}\).
2. \(4x^{6}\cdot x^{-4}=\)\(4x^{2}\).
3. \(a^{-2-3}b^{5-2}=a^{-5}b^{3}=\)\(\dfrac{b^{3}}{a^{5}}\).

Problem 11: Unit-fraction exponents

1. \(\sqrt{25}=\)\(5\) because \(5^{2}=25\).
2. \(\sqrt[3]{64}=\)\(4\) because \(4^{3}=64\).
3. \(\sqrt[4]{81}=\)\(3\) because \(3^{4}=81\).

Problem 12: Radicals to exponents

1. \(7^{1/2}\)
2. The index 3 is the denominator, the power 5 the numerator: \(x^{5/3}\).
3. \(a^{3/4}\)

Problem 13: Exponents to radicals

1. \(\sqrt{x}\)
2. \(\sqrt[3]{y^{2}}\), which equals \((\sqrt[3]{y})^{2}\).
3. \(\sqrt[4]{5^{3}}\), which equals \(\sqrt[4]{125}\).

Problem 14: Rational powers

1. \(8^{2/3}=(\sqrt[3]{8})^{2}=2^{2}=\)\(4\).
2. \(4^{3/2}=(\sqrt{4})^{3}=2^{3}=\)\(8\).
3. \(81^{3/4}=(\sqrt[4]{81})^{3}=3^{3}=\)\(27\).

Problem 15: Negative rational powers

1. \(27^{2/3}=9\), so \(27^{-2/3}=\)\(\dfrac{1}{9}\).
2. \(16^{1/2}=4\), so \(16^{-1/2}=\)\(\dfrac{1}{4}\).
3. \(32^{3/5}=2^{3}=8\), so \(32^{-3/5}=\)\(\dfrac{1}{8}\).

Problem 16: Reading a graph

1. The point (9, 3) is on the graph, so \(9^{1/2}=\)\(3\).
2. 10 is just past 9, so the curve is a little above 3: about 3.2 (the calculator value is 3.16).
3. \(x^{1/2}=4\) means \(x=4^{2}=\)\(16\).

Problem 17: Simplify numeric square roots

1. \(\sqrt{50}=\sqrt{25\cdot2}=\)\(5\sqrt{2}\).
2. The largest perfect square factor of 48 is 16: \(\sqrt{16\cdot3}=\)\(4\sqrt{3}\).
3. \(\sqrt{100\cdot2}=\)\(10\sqrt{2}\).

Problem 18: Rational exponent rules

1. Add: \(x^{1/2+3/2}=x^{2}\), so \(x^{2}\).
2. Subtract: \(x^{5/3-2/3}=x^{1}\), so \(x\).
3. Multiply: \(x^{3/4\cdot2}=x^{3/2}\), so \(x^{3/2}\).
4. \(x^{2/3\cdot6}=x^{4}\), so \(x^{4}\).

Problem 19: Square roots with variables

1. \(x^{6/2}=\)\(x^{3}\).
2. \(\sqrt{9\cdot2\cdot x^{4}}=3x^{2}\sqrt{2}\), so \(3x^{2}\sqrt{2}\).
3. \(75a^{5}=25\cdot3\cdot a^{4}\cdot a\), so the root is \(5a^{2}\sqrt{3a}\).

Problem 20: Square garden

\(s=\sqrt{98}=\sqrt{49\cdot2}=\)\(7\sqrt{2}\text{ ft}\)\(\approx\)\(9.9\text{ ft}\).

Perimeter: \(4\cdot7\sqrt{2}=28\sqrt{2}\approx\)\(39.6\text{ ft}\) of fencing.

Problem 21: Orbit model

1. \(4^{3/2}=(\sqrt{4})^{3}=2^{3}=\)\(8\text{ years}\). \(9^{3/2}=(\sqrt{9})^{3}=3^{3}=\)\(27\text{ years}\). Both match the graph.
2. \(d^{3/2}=64\). Raise both sides to the power \(\frac{2}{3}\): \(d=64^{2/3}=(\sqrt[3]{64})^{2}=4^{2}=\)\(16\text{ AU}\). Check: \(16^{3/2}=4^{3}=64\).

Problem 22: Cube crate

\(s^{3}=512\), so \(s=512^{1/3}=\sqrt[3]{512}=\)\(8\text{ in}\) because \(8^{3}=512\).

A cube has 6 square faces: \(6\cdot8^{2}=6\cdot64=\)\(384\text{ square inches}\).

Problem 23: Patio diagonal

\(d^{2}=6^{2}+6^{2}=72\), so \(d=\sqrt{72}=\sqrt{36\cdot2}=\)\(6\sqrt{2}\text{ ft}\)\(\approx\)\(8.5\text{ ft}\).

Problem 24: Find the errors

1. Jamal did not cube the coefficient. \((2x^{3})^{3}=2^{3}x^{9}=\)\(8x^{9}\).
2. Aisha multiplied the base by the exponent. The denominator 2 means a square root: \(16^{3/2}=(\sqrt{16})^{3}=4^{3}=\)\(64\).

Problem 25: Cube with a variable edge

Edge: \((27x^{6})^{1/3}=27^{1/3}\cdot x^{6/3}=\)\(3x^{2}\). Check: \((3x^{2})^{3}=27x^{6}\).

Surface area: \(6\cdot(3x^{2})^{2}=6\cdot9x^{4}=\)\(54x^{4}\)\(\text{ square units}\).

See every chapter on the Algebra 1 math page.