Learn – Algebra foundations – Radicals

1 Powers and roots

1.1 Exercises

1.1.1 Problem set

Simplify the following.

  1. \sqrt[4]{3^8}
  2. \sqrt[5]{2^{10}}
  3. \sqrt[9]{27^3}
  4. \sqrt[8]{81^4}
  5. \sqrt[6]{8^{10}}
  6. \sqrt[9]{27^{12}}

2 Splitting and combining

2.1 Exercises

2.1.1 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. \sqrt{xyz} = \sqrt{x}\sqrt{y}\sqrt{z}
  2. \sqrt{p+q} = \sqrt{p}+\sqrt{q}
  3. \sqrt{a^2+b^2} = a+b
  4. \sqrt{x^2-y^2} = x-y
  5. \sqrt{x}+\sqrt{x} = 2\sqrt{x}
  6. 8\sqrt{a}-5\sqrt{a} = 3\sqrt{a}
  7. 11\sqrt[3]{p}+5\sqrt[3]{p} = 16\sqrt[3]{p}

2.1.2 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. \sqrt[3]{p^3q^3} = pq
  2. \sqrt[5]{a^5-b^5} = a-b
  3. xy = \sqrt[3]{x^3y^3}
  4. p^2q = \sqrt[5]{p^{10}q^5}
  5. \sqrt[4]{a^8-b^{12}} = a^2-b^3
  6. \sqrt[7]{x^7y^{21}} = xy^3

2.1.3 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. \sqrt{\frac{ab}{cd}} = \frac{\sqrt{a}\sqrt{b}}{\sqrt{c}\sqrt{d}}
  2. \sqrt{\frac{p+q}{r+s}} = \frac{\sqrt{p+q}}{\sqrt{r+s}}
  3. \frac{\sqrt{x-y}}{\sqrt{z+w}} = \sqrt{\frac{x-y}{z+w}}
  4. \sqrt{\frac{a+b}{c+d}} = \frac{\sqrt{a}+\sqrt{b}}{\sqrt{c}+\sqrt{d}}

2.1.4 Problem set

Simplify the following.

  1. \sqrt{a}\sqrt{a}
  2. \sqrt{11}\sqrt{11}
  3. \sqrt[3]{a}\sqrt[3]{a}\sqrt[3]{a}
  4. \sqrt[3]{a^2}\sqrt[3]{a^4}
  5. \frac{\sqrt[3]{a^3b}}{\sqrt[3]{b}}
  6. \frac{\sqrt[3]{a^4}}{\sqrt[3]{a}}

2.1.5 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. x = \sqrt{x}\sqrt{x}
  2. p = \sqrt{p}\sqrt{p}
  3. a = \sqrt[3]{a}\sqrt[3]{a}\sqrt[3]{a}
  4. x = \sqrt[5]{x}\sqrt[5]{x}\sqrt[5]{x}\sqrt[5]{x}
  5. p^2 = \sqrt{p}\sqrt{p^3}
  6. a^5 = \sqrt{a^{13}}\sqrt{a^{12}}
  7. x^4 = \sqrt{x^{11}}\sqrt{x^{5}}

2.1.6 Problem set

Simplify the following.

  1. \frac{7}{\sqrt{7}}
  2. \frac{10}{\sqrt{10}}
  3. \frac{11}{\sqrt{11}}
  4. \frac{8}{\sqrt{2}}
  5. \frac{12}{\sqrt{3}}
  6. \frac{20}{\sqrt{5}}

2.1.7 Problem set

Write each of the following as a simpler radical.

  1. \sqrt{8}
  2. \sqrt{18}
  3. \sqrt{20}
  4. \sqrt{45}

2.1.8 Problem set

Simplify the following.

  1. \frac{-\sqrt{8}}{4}
  2. \frac{-9}{\sqrt{27}}
  3. \frac{-16}{\sqrt{32}}

2.1.9 Problem set

Simplify the following.

  1. \frac{\sqrt{27}}{3}
  2. \frac{\sqrt{50}}{5}
  3. \frac{\sqrt{500}}{10}
  4. \frac{\sqrt[3]{81}}{3}
  5. \frac{\sqrt[4]{32}}{2}

2.1.10 Problem set

Simplify the following.

  1. \sqrt[3]{\frac{27x^6}{y^3}}
  2. \sqrt[3]{\frac{125a^6}{b^9a^{12}}}
  3. \sqrt[5]{-\frac{2^{10}a^2b^2}{a^{12}b^{17}}}
  4. \sqrt[5]{-\frac{2^{10}a^{-2}b^{-2}}{a^{-12}b^{-17}}}
  5. \sqrt{\sqrt{\frac{16x^8}{y^{12}z^4}}}
  6. \sqrt{\frac{\sqrt{16x^8}}{\sqrt{y^{12}z^4}}}

2.1.11 Problem set

Simplify the following.

  1. \sqrt[3]{\frac{200^3}{8}}
  2. \sqrt[6]{\frac{12^3}{3^3}}
  3. \sqrt{2^63^37^2}
  4. \sqrt{2^73^35^3}
  5. \sqrt{3^32^56^3}
  6. \sqrt[3]{3^72^4}
  7. \sqrt[3]{2^53^26^4}

2.1.12 Problem set

Simplify the following.

  1. \frac{50}{\sqrt{125}}
  2. \frac{\sqrt[3]{250}}{20}
  3. \frac{\sqrt{5760}}{80}
  4. \frac{\sqrt{12960}}{18}
  5. \frac{\sqrt[3]{-250}}{5}
  6. \frac{\sqrt[3]{-8640}}{15}
  7. \sqrt[10]{81^2\times 81 \times 81^2}
  8. \sqrt[3]{24^2\times 3}

2.1.13 Problem set

Simplify the following.

  1. \sqrt{6}\sqrt{24}
  2. \sqrt[4]{3}\sqrt[4]{9}\sqrt[4]{3}
  3. \sqrt[3]{3}\sqrt[3]{9}
  4. \sqrt[3]{-125}\times\sqrt[3]{125}
  5. \sqrt[3]{-1000000}\times\sqrt{0.01}
  6. \sqrt[4]{9}\sqrt[4]{81}\sqrt[4]{9}

2.1.14 Problem set

Simplify the following.

  1. \frac{\sqrt{18}}{3}\frac{\sqrt{8}}{2}
  2. \frac{2}{\sqrt{50}}\frac{10}{\sqrt{8}}
  3. \frac{6}{\sqrt[3]{32}}\frac{6}{\sqrt[3]{54}}

2.1.15 Problem set

Simplify the following.

  1. \frac{\sqrt[3]{54}}{\sqrt[3]{250}}
  2. \frac{\sqrt[5]{3}}{\sqrt[3]{5}}\frac{\sqrt[3]{15}}{\sqrt[5]{9}}
  3. \frac{\sqrt[5]{125}}{\sqrt[5]{8}}\frac{\sqrt[10]{64}}{\sqrt[5]{25}}
  4. \frac{\sqrt{3}\sqrt{6} + \sqrt{8}}{10}

3 Solving equations

3.1 Exercises

3.1.1 Problem set

Solve for x. (Assume x \ge 0)

  1. \sqrt{2^{100}} = 2^x
  2. 2^x = (\sqrt{2})^x
  3. (\sqrt{8})^x = (\sqrt{2})^{x+2}
  4. \sqrt[3]{10^{300}} = 10^x
  5. \left(\sqrt[3]{5}\right)^{30} = 5^x
  6. \left(\sqrt[5]{5}\right)^{25} = \left(\sqrt{5}\right)^x
  7. \left(\sqrt[10]{0.1}\right)^{2x} = {0.01}^{x-9}

3.1.2 Problem set

Solve for x.

  1. \sqrt{3}\sqrt{2} = 6^x
  2. \sqrt{2}\sqrt{5} = \sqrt{\frac{x}{4}}
  3. \sqrt{\frac{3}{2}}\sqrt{6} = 27^x
  4. \sqrt[4]{x} = \sqrt{2}\sqrt{10}

3.1.3 Problem set

  1. Solve simultaneously: (x+10)^2 = 81 and x < -10
  2. Solve simultaneously: \sqrt{x^2 - 64} = 6, x < 4
  3. Solve simultaneously: \sqrt{x^2 + 25} = 13, 0 < 2x - 14 < 15
  4. Solve simultaneously: \sqrt{x^2 + 36} = -10, |2x+4| < 4