C2.4; E2.4 - Index laws with letters

C2.4; E2.4 - Index laws with letters

Letters obey the same integer index laws as numbers. You will simplify products, quotients and powers, state when zero is excluded and recognise an unknown exponent by writing both sides with a common base. This lesson covers the shared Core and Extended content.

Products and quotients of algebraic powers

Why does x3×x2x^3\times x^2 give x5x^5, rather than x6x^6? Write the product as (x×x×x)(x×x)(x\times x\times x)(x\times x): there are five factors of xx. The index counts repeated factors, so multiplying powers with the same base adds their indices.

Division reverses that counting. For x≠0x\ne0, x5÷x2=x3x^5\div x^2=x^3, because two factors cancel. Continuing gives x2÷x2=x0=1x^2\div x^2=x^0=1, then x2÷x3=x−1=1xx^2\div x^3=x^{-1}=\frac1x. A negative index means a reciprocal; it does not make the value negative.

Integer index laws

For nonzero aa and integers m,nm,n,

aman=am+naman=am−na0=1a−n=1an\begin{aligned} a^m a^n&=a^{m+n}\\ \frac{a^m}{a^n}&=a^{m-n}\\ a^0&=1\\ a^{-n}&=\frac1{a^n} \end{aligned}

In the last line, take nn positive when using it to interpret a negative power. Positive-integer products also allow a=0a=0; division, zero powers and negative powers require the stated nonzero condition. 000^0 is not defined here. Not given in the List of formulas: learn these laws.

Numerical coefficients multiply or divide in the ordinary way. Apply an index law separately to each letter: different letters are different bases. For example, x2y3x^2y^3 cannot become (xy)5(xy)^5.

Worked example 1: a quotient of products

Question. Simplify 12a5b33a2b\frac{12a^5b^3}{3a^2b}, where a≠0a\ne0 and b≠0b\ne0.

Plan. Divide the coefficients and subtract indices for matching letters.

12a5b33a2b=123a5−2b3−1separate the bases=4a3b2simplify each index\begin{aligned} \frac{12a^5b^3}{3a^2b}&=\frac{12}{3}a^{5-2}b^{3-1} && \text{separate the bases}\\ &=4a^3b^2 && \text{simplify each index} \end{aligned}

Answer. 4a3b24a^3b^2, for the stated nonzero values.

Check. Multiply by the original denominator: 3a2b×4a3b2=12a5b33a^2b\times4a^3b^2=12a^5b^3, recovering the numerator.

Change only the denominator's power of aa, from 22 to 77. Now more factors of aa occur below the line, so some must remain there.

Worked example 2: a negative resulting index

Question. Simplify 12a5b33a7b\frac{12a^5b^3}{3a^7b}, where a≠0a\ne0 and b≠0b\ne0. Give your answer with positive indices.

Plan. Subtract as before, then write the negative power as a reciprocal.

12a5b33a7b=4a5−7b3−1divide matching bases=4a−2b2subtract the indices=4b2a2negative power is reciprocal\begin{aligned} \frac{12a^5b^3}{3a^7b}&=4a^{5-7}b^{3-1} && \text{divide matching bases}\\ &=4a^{-2}b^2 && \text{subtract the indices}\\ &=\frac{4b^2}{a^2} && \text{negative power is reciprocal} \end{aligned}

Answer. 4b2a2\frac{4b^2}{a^2}, retaining a≠0a\ne0 and b≠0b\ne0 from the original expression.

Check. At a=2,b=1a=2,b=1, the original is 12×323×128=1\frac{12\times32}{3\times128}=1; the simplified expression is 44=1\frac44=1.

Products can include negative indices too: 2p−3×5p3=10p0=102p^{-3}\times5p^3=10p^0=10 for p≠0p\ne0. The cancellation leaves the multiplicative identity 11, rather than 00.

Watch out

x3+x2=x5x^3+x^2=x^5 is wrong. Adding indices is a rule for multiplying powers with the same base. The two terms x3x^3 and x2x^2 are unlike terms, so their sum stays as written.

In the next quotient, the denominator contains more factors of the first letter; expect a reciprocal power.

Powers of algebraic expressions

A power outside brackets applies to the whole bracket. In (3x2y)2(3x^2y)^2, two copies of 3x2y3x^2y multiply together. The coefficients give 99; the two copies of x2x^2 give x4x^4; the two copies of yy give y2y^2.

A power of a power or product

For nonzero real a,ba,b and integer m,nm,n,

(am)n=amn(ab)n=anbn(ab)n=anbn\begin{aligned} (a^m)^n&=a^{mn}\\ (ab)^n&=a^n b^n\\ \left(\frac ab\right)^n&=\frac{a^n}{b^n} \end{aligned}

For positive integer powers, zero factors are allowed; a denominator must still be nonzero. Negative powers require the entire base to be nonzero. Not given in the List of formulas: learn these laws.

The reason for multiplying indices is repeated multiplication: (x3)4(x^3)^4 contains four groups of three factors. Brackets matter: (−2x)2=4x2(-2x)^2=4x^2, while −2x2-2x^2 means the square of xx, multiplied by −2-2.

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