Algebra - Indices and Surds

Grade 12IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Laws of Indices: Rules governing the manipulation of powers with the same base.

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Fractional Indices: Expressing roots as powers (e.g., square root as power of 1/2).

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Negative Indices: Expressing reciprocals using negative exponents.

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Simplifying Surds: Factoring out the largest perfect square from under the radical.

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Rationalizing the Denominator: Removing surds from the bottom of a fraction using conjugates.

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Exponential Equations: Solving for an unknown variable located in the exponent by equating bases.

📐Formulae

a^m \times a^n = a^{m+n}

a^m \div a^n = a^{m-n}

(a^m)^n = a^{mn}

a^0 = 1, (a \neq 0)

a^{-n} = \frac{1}{a^n}

a^{\frac{1}{n}} = \sqrt[n]{a}

a^{\frac{m}{n}} = (\sqrt[n]{a})^m

\sqrt{ab} = \sqrt{a} \times \sqrt{b}

\frac{1}{a + \sqrt{b}} = \frac{1}{a + \sqrt{b}} \times \frac{a - \sqrt{b}}{a - \sqrt{b}}

💡Examples

Problem 1:

(8x^6)^{\frac{1}{3}} \times 2x^{-2}

Solution:

4

Explanation:

8^{1/3} \times (x^6)^{1/3} = 2 \times x^2

Problem 2:

\sqrt{75} - \sqrt{12}

Solution:

3\sqrt{3}

Explanation:

\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}

Problem 3:

\frac{4}{3 - \sqrt{5}}

Solution:

3 + \sqrt{5}

Explanation:

(3 + \sqrt{5})

Problem 4:

x

Solution:

x = -8

Explanation:

(3^2)^{x-1} = (3^3)^{x+2}