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

1 Evaluating roots ★★★

Evaluate each expression without a calculator.

1. \(\sqrt{144}\)
2. \(\sqrt[3]{125}\)
3. \(\sqrt[4]{81}\)
4. \(\sqrt[3]{-64}\)
5. \(\sqrt{\dfrac{49}{25}}\)

2 Radical form to exponent form ★★★

Rewrite with a radical sign: a) \(7^{1/2}\) b) \(5^{2/3}\) c) \(x^{3/4}\) d) \((2y)^{1/5}\).

3 Exponent form from radicals ★★★

Rewrite with a rational exponent: a) \(\sqrt{11}\) b) \(\sqrt[3]{a^5}\) c) \(\sqrt[5]{b^2}\) d) \(\dfrac{1}{\sqrt{z}}\).

4 Evaluating rational exponents ★★★

Evaluate: a) \(9^{3/2}\) b) \(8^{4/3}\) c) \(32^{2/5}\) d) \(100^{-1/2}\).

5 Pulling out perfect squares ★★★

Simplify: a) \(\sqrt{50}\) b) \(\sqrt{48}\) c) \(\sqrt{200}\) d) \(\sqrt{75}\).

6 True or false? ★★★

Decide whether each statement is true or false, and justify.

1. \(\sqrt{x^2} = x\) for every real number \(x\).
2. \(\sqrt[3]{-8} = -2\).
3. \(\sqrt{9} + \sqrt{16} = \sqrt{25}\).
4. \(16^{1/2} = 8\).

7 Where is it defined? ★★★

Find all real numbers \(x\) for which each expression is a real number.

1. \(\sqrt{x-5}\)
2. \(\sqrt{7-2x}\)
3. \(\sqrt[3]{x+1}\)
4. \(\sqrt[4]{3x+9}\)

8 How far is the horizon? ★★★

From a lookout whose eyes are \(h\) feet above the water, the distance to the horizon is about \(d = 1.22\sqrt{h}\) miles.

1. Find \(d\) for \(h = 36\) ft (about 11 m).
2. Find \(d\) for \(h = 144\) ft.
3. How high must the lookout be to see \(12.2\) miles?

9 Simplifying with variables ★★★

Simplify (assume all variables are non-negative): a) \(\sqrt{18x^5}\) b) \(\sqrt[3]{40a^4}\) c) \(\sqrt{12}\cdot\sqrt{27}\) d) \(\sqrt{5}\cdot\sqrt{20}\).

10 Rationalizing denominators ★★★

Rationalize and simplify: a) \(\dfrac{6}{\sqrt{8}}\) b) \(\dfrac{4}{\sqrt{7}-\sqrt{3}}\) c) \(\dfrac{3}{\sqrt[3]{2}}\).

11 Exponent rules ★★★

Simplify for \(x, y \gt 0\), and write answers with positive exponents.

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

12 Negative rational exponents ★★★

Evaluate: a) \(\left(\dfrac{27}{8}\right)^{-2/3}\) b) \(81^{-3/4}\) c) \(64^{5/6}\).

13 A square root equation ★★★

Solve \(\sqrt{3x-5} = 4\) and check your answer.

14 A cube root equation ★★★

Solve \(\sqrt[3]{x+4} = -2\).

15 Isolate first ★★★

Solve \(\sqrt{x-3} + 5 = 9\).

16 Domain, range and table ★★★

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

1. Give the domain and the range.
2. Complete a table of values for \(x = -2, -1, 2, 7\).
3. Describe how the graph is obtained from \(y = \sqrt{x}\).

17 An extraneous solution ★★★

Solve \(\sqrt{2x+15} = x\). Identify any extraneous solution.

18 Both candidates survive ★★★

Solve \(\sqrt{5x+1} = x+1\). Is there an extraneous solution?

19 Two radicals ★★★

Solve \(\sqrt{x+6} - \sqrt{x-1} = 1\).

20 Equations with rational exponents ★★★

Solve: a) \(x^{2/3} = 25\) b) \((x+2)^{3/2} = 64\).

21 Find the equation from the graph ★★★

The curve below passes through \(P(-4,-3)\), \(Q(-3,-2)\), \(R(0,-1)\) and \(S(5,0)\). It is a radical function of the form \(f(x) = a\sqrt{x-h}+k\).

1. Find \(h\), \(k\) and \(a\), then write \(f(x)\).
2. Give the domain and range.
3. Solve \(f(x) = 1\).

22 The storage cube ★★★

A cube-shaped storage box holds \(2{,}744\text{ cm}^3\).

1. Find the edge length \(s\).
2. Find the total surface area of the box.
3. Express \(s\) as a power of the volume \(V\).

23 A growing colony ★★★

The number of bacteria in a dish is \(N(t) = 200\cdot 4^{t/2}\), where \(t\) is the time in hours.

1. Compute \(N(1)\), \(N(3)\) and \(N(5)\).
2. After how many hours are there \(3{,}200\) bacteria?
3. Explain why the colony doubles every hour.

24 Find the student’s error ★★★

A student solved \(\sqrt{x-2} = x-4\) like this: “Square both sides: \(x - 2 = x^2 - 8x + 16\). So \(x^2 - 9x + 18 = 0\), i.e. \((x-3)(x-6) = 0\). The solutions are \(x = 3\) and \(x = 6\).” Find and correct the mistake.