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Number - Indices and Standard Form

Grade 9IGCSE

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

🔑Concepts

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Definition of base and index (exponent/power).

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Laws of Indices for multiplication, division, and powers.

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

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Negative Indices: Representing reciprocals using negative exponents.

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Fractional Indices: Relationship between powers and roots (radicals).

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Standard Form (Scientific Notation): Writing numbers in the form A×10nA \times 10^n where 1≤A<101 \le A < 10.

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Converting very large and very small numbers into standard form.

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Performing arithmetic operations (multiplication and division) with numbers in standard form.

📐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

a−n=1ana^{-n} = \frac{1}{a^n}

a1n=ana^{\frac{1}{n}} = \sqrt[n]{a}

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

(ab)n=anbn(ab)^n = a^n b^n

Standard Form: A×10nA \times 10^n (where 1≤A<101 \le A < 10 and nn is an integer)

💡Examples

Problem 1:

Simplify (3x2y4)×(4x5y−2)(3x^2y^4) \times (4x^5y^{-2}).

Solution:

12x7y212x^7y^2

Explanation:

Multiply the coefficients (3×4=123 \times 4 = 12). Add the powers for xx (2+5=72 + 5 = 7) and add the powers for yy (4+(−2)=24 + (-2) = 2).

Problem 2:

Evaluate 27−2327^{-\frac{2}{3}} without using a calculator.

Solution:

19\frac{1}{9}

Explanation:

First, handle the negative index by taking the reciprocal: 1/272/31 / 27^{2/3}. Next, apply the fractional index: 271/327^{1/3} is the cube root of 27, which is 3. Finally, square the result: 32=93^2 = 9. Thus, the answer is 1/91/9.

Problem 3:

Calculate (4×105)×(5×107)(4 \times 10^5) \times (5 \times 10^7), giving your answer in standard form.

Solution:

2×10132 \times 10^{13}

Explanation:

Multiply the numbers: 4×5=204 \times 5 = 20. Add the powers of 10: 105×107=101210^5 \times 10^7 = 10^{12}. This gives 20×101220 \times 10^{12}. To convert to standard form (1≤A<101 \le A < 10), rewrite 20 as 2×1012 \times 10^1. Then 2×101×1012=2×10132 \times 10^1 \times 10^{12} = 2 \times 10^{13}.