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Number - Indices and Standard Form

Grade 10IGCSE

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 of powers.

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Meaning of zero, negative, and fractional indices.

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Standard Form (Scientific Notation) expressed as a×10na \times 10^n, where 1≤a<101 \leq a < 10 and nn is an integer.

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Converting between ordinary decimal numbers and standard form.

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Performing arithmetic operations (addition, subtraction, multiplication, 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 (for a≠0a \neq 0)

a−n=1ana^{-n} = \frac{1}{a^n}

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

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

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

💡Examples

Problem 1:

Simplify (3x2y−3)3(3x^2 y^{-3})^3 leaving your answer with positive indices.

Solution:

27x6y9\frac{27x^6}{y^9}

Explanation:

Apply the power to each term inside the bracket: 33⋅(x2)3⋅(y−3)3=27x6y−93^3 \cdot (x^2)^3 \cdot (y^{-3})^3 = 27x^6 y^{-9}. To write with positive indices, use the rule a−n=1ana^{-n} = \frac{1}{a^n} to move y−9y^{-9} to the denominator.

Problem 2:

Evaluate 125−23125^{-\frac{2}{3}}.

Solution:

125\frac{1}{25}

Explanation:

First, handle the negative index: 125−23=11252/3125^{-\frac{2}{3}} = \frac{1}{125^{2/3}}. Then, apply the fractional index rule: 1252/3=(1253)2=52=25125^{2/3} = (\sqrt[3]{125})^2 = 5^2 = 25. Thus, the result is 125\frac{1}{25}.

Problem 3:

Calculate (4.5×107)×(2×10−3)(4.5 \times 10^7) \times (2 \times 10^{-3}), giving your answer in standard form.

Solution:

9×1049 \times 10^4

Explanation:

Multiply the coefficients (4.5×2=94.5 \times 2 = 9) and use the index law for the powers of 10 (107×10−3=107−3=10410^7 \times 10^{-3} = 10^{7-3} = 10^4). Since 99 is between 1 and 10, the answer is already in standard form.

Problem 4:

Work out (3.2×105)+(6×104)(3.2 \times 10^5) + (6 \times 10^4), giving your answer in standard form.

Solution:

3.8×1053.8 \times 10^5

Explanation:

To add, the powers of 10 must be the same. Convert 6×1046 \times 10^4 to 0.6×1050.6 \times 10^5. Then add the coefficients: (3.2+0.6)×105=3.8×105(3.2 + 0.6) \times 10^5 = 3.8 \times 10^5.