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Algebra - Algebraic Indices

Grade 9IGCSE

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

๐Ÿ”‘Concepts

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An index (also called a power or exponent) tells us how many times a base number is multiplied by itself. In the expression ana^n, aa is the base and nn is the index.

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Multiplication Law: When multiplying terms with the same base, add the indices: amร—an=am+na^m \times a^n = a^{m+n}.

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Division Law: When dividing terms with the same base, subtract the indices: amรทan=amโˆ’na^m \div a^n = a^{m-n}.

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Power of a Power Law: When raising a power to another power, multiply the indices: (am)n=amn(a^m)^n = a^{mn}.

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Zero Index: Any non-zero base raised to the power of zero is equal to 1: a0=1a^0 = 1 where aโ‰ 0a \neq 0.

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Negative Indices: A negative index indicates a reciprocal: aโˆ’n=1ana^{-n} = \frac{1}{a^n}.

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Fractional Indices: The denominator of a fractional index represents the root, and the numerator represents the power: amn=amn=(an)ma^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m.

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Power of a Product/Quotient: Indices apply to every factor inside the bracket: (ab)n=anbn(ab)^n = a^n b^n and (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}.

๐Ÿ“Formulae

amร—an=am+na^m \times a^n = a^{m+n}

aman=amโˆ’n\frac{a^m}{a^n} = a^{m-n}

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

a0=1a^0 = 1

aโˆ’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

๐Ÿ’กExamples

Problem 1:

Simplify the expression: 3x2y3ร—4x5yโˆ’13x^2 y^3 \times 4x^5 y^{-1}

Solution:

3x2y3ร—4x5yโˆ’13x^2 y^3 \times 4x^5 y^{-1} =(3ร—4)ร—x2+5ร—y3+(โˆ’1)= (3 \times 4) \times x^{2+5} \times y^{3 + (-1)} =12x7y2= 12x^7 y^2

Explanation:

Multiply the coefficients (33 and 44) normally. For variables with the same base, use the multiplication law of indices by adding the exponents: 2+5=72+5=7 for xx and 3+(โˆ’1)=23+(-1)=2 for yy.

Problem 2:

Evaluate: 27โˆ’2327^{-\frac{2}{3}}

Solution:

27โˆ’23=1272327^{-\frac{2}{3}} = \frac{1}{27^{\frac{2}{3}}} =1(273)2= \frac{1}{(\sqrt[3]{27})^2} =132= \frac{1}{3^2} =19= \frac{1}{9}

Explanation:

First, convert the negative index into a reciprocal. Then, interpret the fractional index 23\frac{2}{3} as taking the cube root of the base and then squaring the result. The cube root of 2727 is 33, and 33 squared is 99.

Problem 3:

Simplify: (2a3b)48a10b2\frac{(2a^3 b)^4}{8a^{10} b^2}

Solution:

(2a3b)48a10b2\frac{(2a^3 b)^4}{8a^{10} b^2} =24(a3)4b48a10b2= \frac{2^4 (a^3)^4 b^4}{8a^{10} b^2} =16a12b48a10b2= \frac{16 a^{12} b^4}{8a^{10} b^2} =2a12โˆ’10b4โˆ’2= 2 a^{12-10} b^{4-2} =2a2b2= 2a^2 b^2

Explanation:

First, expand the brackets in the numerator using the power of a power law. (2a3b)4(2a^3 b)^4 becomes 16a12b416a^{12}b^4. Then, divide the coefficients and subtract the indices for aa and bb using the division law.