Radical Expressions - Questions with Solutions for Grade 10

Grade 10 questions on how to use important formulas to simplify radical algebraic expressions are presented below. Review the formulas and examples, try solving the practice questions, and click the arrows to view the step-by-step solutions.

Important Formulas & Examples

Formula A

If $n$ and $m$ are positive integers and $\sqrt[n]{y}$ is a real number, then:

$$ \left( \sqrt[n]{y} \right)^m = \sqrt[n]{y^m} $$

Examples:

  1. Since $\sqrt{5}$ is a real number:
    $$(\sqrt{5})^2 = \sqrt{5^2} = 5$$
  2. Since $\sqrt[3]{-7}$ is a real number:
    $$(\sqrt[3]{-7})^6 = \sqrt[3]{(-7)^6} = \sqrt[3]{(-1)^6 \cdot 7^6} = \sqrt[3]{(7^2)^3} = 7^2 = 49$$

Formula B

If $n$ is an EVEN positive integer, then:

$$ \sqrt[n]{y^n} = |y| $$

Examples:

  1. $\sqrt{16} = \sqrt{4^2} = |4| = 4$
  2. $\sqrt[4]{\left( -3 \right)^4} = |-3| = 3$
  3. $\sqrt{(x-2)^2} = |x-2|$
  4. $\sqrt[4]{x^4} = |x|$
  5. $\sqrt{x^4} = \sqrt{(x^2)^2} = |x^2| = x^2$

Formula C

If $n$ is an ODD positive integer, then:

$$ \sqrt[n]{y^n} = y $$

Examples:

  1. $\sqrt[3]{-1} = \sqrt[3]{(-1)^3} = -1$
  2. $\sqrt[5]{(-2)^5} = -2$
  3. $\sqrt[3]{-27} = \sqrt[3]{(-3)^3} = -3$
  4. $\sqrt[5]{x^5} = x$
  5. $\sqrt[3]{-x^6} = \sqrt[3]{(-x^2)^3} = -x^2$

Practice Questions With Answers

Rewrite, if possible, the following expressions without radicals (simplify):

Question 1

Simplify: $\left( \sqrt[3]{x} \right)^3$

View Solution

The index of the radical $3$ is odd and equal to the power of the radicand.

$$ \left( \sqrt[3]{x} \right)^3 = x $$

Question 2

Simplify: $\left( \sqrt{x} \right)^2$

View Solution

Since $\sqrt{x}$ is a real number, $x$ must be positive or zero, and therefore $|x| = x$.

$$ \left( \sqrt{x} \right)^2 = \sqrt{x^2} = |x| = x $$

Question 3

Simplify: $-\left( \sqrt{x} \right)^4$

View Solution

$$ -\left( \sqrt{x} \right)^4 = - \sqrt{x^4} = - |x^2| = -x^2 $$

Question 4

Simplify: $\sqrt{-x^2 - 1}$

View Solution

Since $-x^2 - 1$ is always a negative number, $\sqrt{-x^2 - 1}$ is not a real number.

Question 5

Simplify: $\sqrt[8]{x^8}$

View Solution

The index $8$ is even and equal to the power of the radicand.

$$ \sqrt[8]{x^8} = |x| $$

Question 6

Simplify: $\sqrt{x^6}$

View Solution

$$ \sqrt{x^6} = \sqrt{(x^3)^2} = |x^3| $$

Question 7

Simplify: $\sqrt{x \cdot |x|}$

View Solution

If $x < 0$, then $|x| = -x$ and $\sqrt{x \cdot |x|} = \sqrt{-x^2}$ which is not a real number.

If $x \geq 0$, then $|x| = x$ and $\sqrt{x \cdot |x|} = \sqrt{x^2} = |x| = x$.

Question 8

Simplify: $\sqrt[10]{x^{10}}$

View Solution

The index $10$ of the radical is even and equal to the power of the radicand.

$$ \sqrt[10]{x^{10}} = |x| $$

Question 9

Simplify: $\sqrt[3]{(x - 2)^3}$

View Solution

The index $3$ of the radical is odd and equal to the power of the radicand.

$$ \sqrt[3]{(x - 2)^3} = x - 2 $$

Question 10

Simplify: $\sqrt{\frac{x^2}{9}}$

View Solution

$$ \sqrt{\frac{x^2}{9}} = \sqrt{\left(\frac{x}{3}\right)^2} = \left|\frac{x}{3}\right| = \frac{|x|}{3} $$

Question 11

Simplify: $\sqrt[5]{\frac{x^5}{32}}$

View Solution

$$ \sqrt[5]{\frac{x^5}{32}} = \sqrt[5]{\left(\frac{x}{2}\right)^5} = \frac{x}{2} $$

Question 12

Simplify: $\sqrt{(-x + 3)^2}$

View Solution

The radical has an even index and power of radicand.

$$ \sqrt{(-x+3)^2} = | -x + 3 | $$

Question 13

Simplify: $\sqrt{x^2 + 4x + 4}$

View Solution

The radical has an even index and power of radicand.

$$ \sqrt{x^2 + 4x + 4} = \sqrt{(x+2)^2} = |x+2| $$

Links and References