Algebra 1 Exponent Rules and Radicals Test Solutions and Scoring (PDF)

a) \(2^{5 + (-2)} = 2^{3} = 8\).

b) \(\frac{3^{6}}{3^{4}} = 3^{2} = 9\).

8; 9

Scoring: 1 point for each part.

Square every factor: \(4^{2} = 16\), \((x^{3})^{2} = x^{6}\), \((y^{2})^{2} = y^{4}\).

\(16x^{6}y^{4}\)

Scoring: 1 point for the coefficient 16; 1 point for the exponents.

Subtract exponents: \(x^{-3-2} = x^{-5}\) and \(y^{4-1} = y^{3}\).

\(x^{-5}\) belongs in the denominator.

\(\frac{y^{3}}{x^{5}}\)

Scoring: 1 point for the exponents; 1 point for positive exponents.

\(\sqrt{98} = \sqrt{49 \cdot 2} = 7\sqrt{2}\).

\(7\sqrt{2}\) inches

Scoring: 1 point for factoring out 49; 1 point for the simplified form.

a) \(\sqrt[3]{64} = 4\).

b) \((\sqrt{9})^{3} = 3^{3} = 27\).

4; 27

Scoring: 1 point for each part.

a) \(\sqrt[4]{x^{3}}\).

b) \(y^{2/5}\).

Scoring: 1 point for each part.

a) \(\sqrt{81} = 9\) is rational.

b) 11 is not a perfect square, so \(\sqrt{11}\) is irrational.

c) If \(3 + \sqrt{5}\) were rational, subtracting 3 would make \(\sqrt{5}\) rational, which is false.

rational; irrational; irrational

Scoring: 1 point per part.

a) \(50 \cdot 3^{0} = 50\).

b) \(50 \cdot 3^{4} = 50 \cdot 81 = 4050\).

c) \(50 \cdot 3^{t} = 1350\), so \(3^{t} = 27 = 3^{3}\) and \(t = 3\).

50; 4,050; 3 hours

Scoring: 1 point per part.

The power applies to both factors, 2 and \(x\). The student cubed only \(x\).

\((2x)^{3} = 2^{3}x^{3} = 8x^{3}\).

\(8x^{3}\)

Scoring: 1 point for the explanation; 1 point for the correct result.