Multiply Radicals - Questions with Solutions for Grade 10

Grade 10 questions on how to multiply expressions with radicals including their solutions are presented.

Product (Multiplication) of Radicals With the Same Index

Multiplication formula of radicals with equal indices is given by
\[ \Large{\color{red}{\sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{x \cdot y}}} \]

Examples

  1. \(\sqrt{3} \cdot \sqrt{12} = \sqrt{3 \cdot 12} = \sqrt{36} = 6\).
  2. \(\sqrt[3]{27x^3} = \sqrt[3]{27} \cdot \sqrt[3]{x^3} = 3x\).
  3. \(\sqrt[4]{\frac{1}{12}} \cdot \sqrt[4]{3} \cdot \sqrt[4]{64} = \sqrt[4]{\frac{1}{12} \cdot 3 \cdot 64} = \sqrt[4]{16} = 2\).
  4. \(\color{black}{3\sqrt[10]{x^3} \cdot 5\sqrt[10]{x^5} \cdot \sqrt[10]{x^2} = (3 \cdot 5) \sqrt[10]{x^3 \cdot x^5 \cdot x^2} = 15 \sqrt[10]{x^{10}} = 15|x| = 15x}\) (assuming \(x \geq 0\)).

Questions With Answers

Use the above multiplication formula to simplify the following expressions
  1. \( 4\sqrt{2} \cdot 7\sqrt{32} \)
  2. \( 6\sqrt{x} \cdot \frac{2}{3}\sqrt{x} \)
  3. \( 2\sqrt[3]{\frac{1}{32}} \cdot \sqrt[3]{128} \cdot \sqrt[3]{16} \)
  4. \( -8\sqrt[5]{x^2} \cdot 2\sqrt[5]{x^3} \)
  5. \( \sqrt{(x-3)} \cdot \sqrt{(x-3)} \)
  6. \( \sqrt[8]{x} \cdot 5\sqrt[8]{x^4} \cdot 2\sqrt[8]{x^3} \)

Solutions to the Above Problems

  1. \( \quad 4\sqrt{2} \cdot 7\sqrt{32} = (4 \cdot 7)\sqrt{2 \cdot 32} = 28\sqrt{64} = 28 \cdot 8 = 224 \)
  2. \( \quad 6\sqrt{x} \cdot \frac{2}{3}\sqrt{x} = \left(6 \cdot \frac{2}{3}\right)\sqrt{x \cdot x} = 4\sqrt{x^2} = 4|x| = 4x \quad (\text{assuming } x \geq 0) \)
  3. \( \quad 2\sqrt[3]{\frac{1}{32}} \cdot \sqrt[3]{128} \cdot \sqrt[3]{16} = 2\sqrt[3]{\frac{1}{32} \cdot 128 \cdot 16} = 2\sqrt[3]{64} = 2 \cdot 4 = 8 \)
  4. \( \quad -8\sqrt[5]{x^2} \cdot 2\sqrt[5]{x^3} = (-8 \cdot 2)\sqrt[5]{x^2 \cdot x^3} = -16\sqrt[5]{x^5} = -16x \)
  5. \( \quad \sqrt{(x-3)} \cdot \sqrt{(x-3)} = \sqrt{(x-3)(x-3)} = \sqrt{(x-3)^2} = |x-3| = x-3 \) (Note: \(\sqrt{(x-3)}\) is a real number and therefore \((x-3) \geq 0\) which leads to \(|x-3| = x-3\))
  6. \( \quad \sqrt[8]{x} \cdot 5\sqrt[8]{x^4} \cdot 2\sqrt[8]{x^3} = (5 \cdot 2)\sqrt[8]{x \cdot x^4 \cdot x^3} = 10\sqrt[8]{x^8} = 10|x| = 10x \) (Assuming \(x \geq 0\))

References and Links