Radicals and Rational Exponents: math test, Grade 11 – download the PDF

1 Roots and exponents / 3 pts

Evaluate without a calculator (0.5 point each): a) \(\sqrt{169}\) b) \(\sqrt[3]{-216}\) c) \(625^{1/4}\) d) \(49^{3/2}\) e) \(8^{-2/3}\) f) \(\left(\dfrac{16}{81}\right)^{3/4}\).

2 Simplifying radicals / 4 pts

Simplify (variables are positive): a) \(\sqrt{98}\) (1 pt) b) \(\sqrt[3]{54x^4}\) (1.5 pt) c) \(\dfrac{10}{\sqrt{5}}\) (0.5 pt) d) \(\dfrac{5}{3-\sqrt{2}}\) (1 pt).

3 Exponent rules / 3 pts

Simplify, with positive exponents (\(x, y \gt 0\)): a) \(x^{5/6}\cdot x^{1/3}\) b) \((8a^6)^{2/3}\) c) \(\left(x^{-1/2}y^{3/4}\right)^4\).

4 Solving equations / 4 pts

Solve and check: a) \(\sqrt{4x+1} = 7\) (1.5 pt) b) \(\sqrt[3]{2x-5} + 1 = 4\) (1.5 pt) c) \((x-3)^{3/2} = 125\) (1 pt).

5 Extraneous solution / 3 pts

Solve \(\sqrt{3x+7} = x+1\). Show every step and identify any extraneous solution.

6 A radical function / 3 pts

Let \(f(x) = 2\sqrt{x+3} - 4\).

1. Give the domain and the range. (1 pt)
2. Complete a table for \(x = -3, -2, 1, 6\). (1 pt)
3. Find the \(x\)-intercept and the exact \(y\)-intercept. (1 pt)