Surds and Indices: Notes and 100 Practice Questions

Surds and Indices: Essential Notes

What are Surds?

A surd is an irrational root of a rational number that cannot be simplified to remove the root sign (√) completely.
Examples: $\sqrt{2}$ , $\sqrt{5}$ , $\sqrt{7}$ are surds. But $\sqrt{4} = 2$ is not a surd.

Key Properties of Surds

$\sqrt{a} \times \sqrt{b} = \sqrt{a b}$
$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
$(\sqrt{a})^{2} = a$
To rationalize denominators: Multiply numerator and denominator by a suitable surd to remove the root from the denominator.

What are Indices?

Indices (or exponents/powers) denote repeated multiplication of a number.
Example: $2^{4} = 2 \times 2 \times 2 \times 2 = 16$ .

Key Laws of Indices

$a^{m} \times a^{n} = a^{m + n}$
$\frac{a^{m}}{a^{n}} = a^{m - n}$
$(a^{m})^{n} = a^{m n}$
$a^{0} = 1$ (where $a \neq 0$ )
$a^{- n} = \frac{1}{a^{n}}$
$a^{1 / n} = \sqrt[n]{a}$

Simplification Tips

Convert all roots to fractional indices for easy calculation: $\sqrt{a} = a^{1 / 2}$ , $\sqrt{a} = a^{1 / 3}$
Always use index laws to simplify expressions before calculation.
For surds, try to express all terms under a common root if possible.

Examples

Example 1: Simplify $\sqrt{12} + \sqrt{27}$
$\sqrt{12} = 2 \sqrt{3}$ , $\sqrt{27} = 3 \sqrt{3}$
Sum: $2 \sqrt{3} + 3 \sqrt{3} = 5 \sqrt{3}$

Example 2: Simplify $2^{3} \times 2^{4}$
$2^{3 + 4} = 2^{7} = 128$

Example 3: Simplify $\frac{3^{5}}{3^{2}}$
$3^{5 - 2} = 3^{3} = 27$

Example 4: Rationalize $\frac{1}{\sqrt{2}}$
Multiply numerator and denominator by $\sqrt{2}$ :
$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Common Mistakes to Avoid

Adding unlike surds (e.g., $\sqrt{2} + \sqrt{3}$ cannot be simplified).
Forgetting index laws or misapplying them.
Not rationalizing denominators when required.
Ignoring negative or zero exponents.

100 Practice Questions on Surds and Indices

Questions 1–20 (with solutions)

Simplify: $\sqrt{18}$
Ans: $3 \sqrt{2}$
Simplify: $\sqrt{50} + \sqrt{8}$
Ans: $5 \sqrt{2} + 2 \sqrt{2} = 7 \sqrt{2}$
Simplify: $2^{3} \times 2^{2}$
Ans: $2^{5} = 32$
Simplify: $9^{1 / 2}$
Ans: $3$
Simplify: $16^{3 / 4}$
Ans: $(16^{1 / 4})^{3} = (2)^{3} = 8$
Simplify: $\frac{1}{\sqrt{5}}$ (rationalize)
Ans: $\frac{\sqrt{5}}{5}$
Simplify: $3^{- 2}$
Ans: $\frac{1}{9}$
Simplify: $\sqrt{32} - \sqrt{8}$
Ans: $4 \sqrt{2} - 2 \sqrt{2} = 2 \sqrt{2}$
Simplify: $5^{0} + 2^{0}$
Ans: $1 + 1 = 2$
Simplify: $\frac{2^{5}}{2^{3}}$
Ans: $2^{2} = 4$
Simplify: $\sqrt{45} + \sqrt{20}$
Ans: $3 \sqrt{5} + 2 \sqrt{5} = 5 \sqrt{5}$
Simplify: $8^{2 / 3}$
Ans: $(8^{1 / 3})^{2} = (2)^{2} = 4$
Simplify: $\sqrt{27} - \sqrt{12}$
Ans: $3 \sqrt{3} - 2 \sqrt{3} = \sqrt{3}$
Simplify: $4^{1.5}$
Ans: $4^{3 / 2} = (2^{2})^{3 / 2} = 2^{3} = 8$
Simplify: $2^{3} \times 4^{2}$
Ans: $2^{3} \times (2^{2})^{2} = 2^{3} \times 2^{4} = 2^{7} = 128$
Simplify: $\frac{1}{\sqrt{3} + 1}$ (rationalize)
Ans: $\frac{\sqrt{3} - 1}{2}$
Simplify: $27^{1 / 3} + 125^{1 / 3}$
Ans: $3 + 5 = 8$
Simplify: $81^{1 / 4}$
Ans: $3$
Simplify: $2^{- 3}$
Ans: $\frac{1}{8}$
Simplify: $\frac{9^{3 / 2}}{3^{2}}$
Ans: $27 / 9 = 3$

Questions 21–100 (Practice Only)

Simplify: $\sqrt{72}$
Simplify: $32^{1 / 5}$
Simplify: $2^{4} \times 2^{- 2}$
Simplify: $49^{1 / 2}$
Simplify: $64^{2 / 3}$
Rationalize: $\frac{1}{\sqrt{7}}$
Simplify: $4^{- 1}$
Simplify: $\sqrt{18} + \sqrt{50}$
Simplify: $3^{0} + 4^{0}$
Simplify: $\frac{5^{4}}{5^{2}}$
Simplify: $\sqrt{75} - \sqrt{12}$
Simplify: $16^{1 / 2} + 81^{1 / 4}$
Simplify: $2^{2} \times 8^{1 / 3}$
Simplify: $5^{- 2}$
Simplify: $100^{1 / 2}$
Simplify: $27^{2 / 3}$
Rationalize: $\frac{1}{2 + \sqrt{3}}$
Simplify: $2^{5} \div 2^{2}$
Simplify: $9^{0} \times 7^{0}$
Simplify: $\sqrt{242}$
Simplify: $3^{3} \times 3^{- 1}$
Simplify: $81^{1 / 2}$
Simplify: $64^{1 / 3}$
Simplify: $2^{3} + 4^{1.5}$
Simplify: $\frac{1}{\sqrt{5} - 2}$ (rationalize)
Simplify: $8^{1 / 3} + 27^{1 / 3}$
Simplify: $16^{- 1 / 2}$
Simplify: $5^{3} \div 5^{5}$
Simplify: $121^{1 / 2}$
Simplify: $2^{6} \times 4^{- 1}$
Simplify: $3^{4} \div 9^{2}$
Simplify: $25^{1 / 2} + 36^{1 / 2}$
Simplify: $\sqrt{32} + \sqrt{18}$
Simplify: $2^{3} \times 8^{- 1}$
Simplify: $49^{3 / 2}$
Rationalize: $\frac{1}{\sqrt{2} + 1}$
Simplify: $81^{0}$
Simplify: $3^{5} \div 3^{2}$
Simplify: $16^{1 / 4}$
Simplify: $64^{1 / 6}$
Simplify: $2^{4} \times 4^{0.5}$
Simplify: $9^{1.5}$
Simplify: $5^{0} + 2^{- 1}$
Simplify: $100^{1 / 2} - 25^{1 / 2}$
Simplify: $8^{2 / 3}$
Simplify: $3^{- 3}$
Simplify: $27^{1 / 3} \times 9^{1 / 2}$
Rationalize: $\frac{1}{\sqrt{3} - 1}$
Simplify: $2^{10} \div 2^{7}$
Simplify: $5^{2} \times 25^{- 1}$
Simplify: $49^{1 / 2} + 64^{1 / 2}$
Simplify: $\sqrt{50} - \sqrt{8}$
Simplify: $4^{2.5}$
Simplify: $3^{2} \times 9^{- 1}$
Simplify: $16^{3 / 4}$
Simplify: $7^{0}$
Simplify: $32^{1 / 5} \times 8^{1 / 3}$
Simplify: $2^{- 2} + 4^{- 1}$
Simplify: $\sqrt{27} + \sqrt{48}$
Simplify: $5^{2} \div 5^{4}$
Simplify: $9^{3 / 2}$
Simplify: $2^{6} \times 8^{- 2}$
Simplify: $81^{1 / 4}$
Simplify: $3^{0} + 2^{0}$
Simplify: $64^{1 / 2} - 16^{1 / 4}$
Simplify: $2^{3} \div 8^{1 / 3}$
Rationalize: $\frac{1}{\sqrt{7} + 2}$
Simplify: $4^{1.5} + 8^{2 / 3}$
Simplify: $25^{- 1 / 2}$
Simplify: $2^{2} \times 4^{1 / 2}$
Simplify: $3^{4} \times 3^{- 2}$
Simplify: $16^{0}$
Simplify: $5^{3} \div 25^{1.5}$
Simplify: $9^{1 / 2} + 16^{1 / 2}$
Simplify: $2^{4} \times 8^{- 1}$
Simplify: $27^{2 / 3}$
Simplify: $4^{- 2}$
Simplify: $3^{2} \div 9^{1 / 2}$
Simplify: $16^{1 / 4} \times 8^{1 / 3}$
Simplify: $2^{0} + 3^{0} + 4^{0}$

Tip:
Practice these questions regularly and always apply the rules and properties of surds and indices for accurate and quick solutions1.

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