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Number - Powers, Roots, and Laws of Indices

Grade 8IGCSE

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

🔑Concepts

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Base and Index: In the expression ana^n, aa is the base and nn is the index (or exponent/power).

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Powers and Roots: Roots are the inverse operation of powers (e.g., x\sqrt{x} is the inverse of x2x^2).

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Multiplication Law: When multiplying terms with the same base, add the indices.

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Division Law: When dividing terms with the same base, subtract the indices.

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Power of a Power: When raising a power to another power, multiply the indices.

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Zero Index: Any non-zero number raised to the power of zero is always 1.

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Negative Indices: A negative index indicates a reciprocal (flipping the base).

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Fractional Indices: The denominator of a fractional index indicates the root, while the numerator indicates the power.

📐Formulae

am×an=am+na^m \times a^n = a^{m+n}

am÷an=am−na^m \div a^n = a^{m-n}

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

a0=1a^0 = 1 (for a≠0a \neq 0)

a−n=1ana^{-n} = \frac{1}{a^n}

a1/n=ana^{1/n} = \sqrt[n]{a}

am/n=(an)ma^{m/n} = (\sqrt[n]{a})^m

💡Examples

Problem 1:

Simplify 54×53÷555^4 \times 5^3 \div 5^5.

Solution:

52=255^2 = 25

Explanation:

Using the multiplication law, 54×53=54+3=575^4 \times 5^3 = 5^{4+3} = 5^7. Then using the division law, 57÷55=57−5=525^7 \div 5^5 = 5^{7-5} = 5^2.

Problem 2:

Evaluate 27−1/327^{-1/3}.

Solution:

1/31/3

Explanation:

First, address the negative index by taking the reciprocal: 1/(271/3)1/(27^{1/3}). The fractional index 1/31/3 means the cube root, so 1/273=1/31/\sqrt[3]{27} = 1/3.

Problem 3:

Simplify (3x2)3(3x^2)^3.

Solution:

27x627x^6

Explanation:

Apply the power to both the coefficient and the variable inside the bracket. 33=273^3 = 27 and (x2)3=x2×3=x6(x^2)^3 = x^{2 \times 3} = x^6.

Problem 4:

Find the value of (1/4)−2(1/4)^{-2}.

Solution:

16

Explanation:

A negative power on a fraction flips the fraction: (4/1)2(4/1)^2. Then 42=164^2 = 16.