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

Question 1: Powers of 3 (2 points)

1. Dividing 3 by 3 gives \(3^{0}=\)\(1\).
2. Dividing \(\frac{1}{3}\) by 3 gives \(3^{-2}=\)\(\dfrac{1}{9}\).

Scoring: 1 point for each correct value.

Question 2: Monomial products and quotients (2 points)

1. \((4)(-2)=-8\), \(x^{5}\), \(y^{5}\): \(-8x^{5}y^{5}\).
2. \(3a^{5}b\): \(3a^{5}b\).

Scoring: 1 point for each correct simplified expression.

Question 3: Combining power rules (2 points)

1. \(x^{-8}\cdot x^{11}=\)\(x^{3}\).
2. \(2^{3}x^{-6}y^{3}=8x^{-6}y^{3}=\)\(\dfrac{8y^{3}}{x^{6}}\).

Scoring: 1 point for each correct answer. Part (b) needs positive exponents.

Question 4: Rational exponent values (2 points)

1. \((\sqrt{25})^{3}=5^{3}=\)\(125\).
2. \(32^{2/5}=(\sqrt[5]{32})^{2}=2^{2}=4\), so \(32^{-2/5}=\)\(\dfrac{1}{4}\).

Scoring: 1 point for each correct value; in part (b), 0 points for the answer 4 or -4.

Question 5: Radical and exponent forms (2 points)

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

Scoring: 1 point for each correct form. Accept any equivalent radical in part (b).

Question 6: Simplify square roots (3 points)

1. \(\sqrt{9\cdot5}=\)\(3\sqrt{5}\).
2. \(\sqrt{16\cdot5}=\)\(4\sqrt{5}\).
3. \(12x^{5}=4\cdot3\cdot x^{4}\cdot x\), so \(2x^{2}\sqrt{3x}\).

Scoring: 1 point for each fully simplified answer. No credit if a perfect square factor remains under the root.

Question 7: Exponent rules with fractions (2 points)

1. \(x^{3/4\cdot8}=\)\(x^{6}\).
2. \(x^{7/3-1/3}=x^{2}\): \(x^{2}\).

Scoring: 1 point for each correct power of \(x\).

Question 8: Graph of a rational power (3 points)

1. The graph passes through (16, 4), so \(y=\)\(4\).
2. \(y=\sqrt{x}\)
3. Yes. \(25^{1/2}=\sqrt{25}=5\), so (25, 5) satisfies the equation.

Scoring: 1 point for the value, 1 point for the radical form, 1 point for substituting 25 and getting 5.

Question 9: Square frame (2 points)

\(s=\sqrt{150}=\sqrt{25\cdot6}=\)\(5\sqrt{6}\text{ in}\). Since \(\sqrt{6}\approx2.449\), \(s\approx\)\(12.2\text{ in}\).

Scoring: 1 point for the simplified radical, 1 point for the decimal.

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