C2.4; E2.4 - Algebraic index laws

C2.4; E2.4 - Algebraic index laws

Does x3×x2x^3\times x^2 mean x5x^5 or x6x^6? Counting equal factors explains the answer and helps you keep coefficients, signs and letters in their correct roles. This lesson develops the integer algebraic index laws in C2.4 and E2.4 and solves equations by rewriting both sides with a common base.

Equal bases

The product x3×x2x^3\times x^2 contains three copies of xx followed by two more: five copies altogether. Hence it is x5x^5. The same laws you used with numerical bases work with variable bases, whenever the expressions are defined.

Equal-base index laws

For a non-zero base aa and integer indices m,nm,n,

am×an=am+nam÷an=am−na0=1,a−n=1an(n>0).\begin{aligned} a^m\times a^n&=a^{m+n}\\ a^m\div a^n&=a^{m-n}\\ a^0&=1,\qquad a^{-n}=\frac1{a^n}\quad(n>0). \end{aligned}

Only powers of the same base combine this way. The non-zero condition is needed for division and for zero or negative powers; a positive integer power of zero itself is allowed.

Not on the supplied formula lists: learn these laws.

Simplify numerical coefficients separately from variable powers. For example, 3x2×4x53x^2\times4x^5 has coefficient 3×4=123\times4=12 and variable part x2+5x^{2+5}, so it is 12x712x^7.

Worked example 1: several equal bases in a product

Question. Simplify 6x4y3×2x−2y6x^4y^3\times2x^{-2}y, for non-zero x,yx,y. Give positive indices.

Plan. Multiply the coefficients and add the indices of each matching variable.

6x4y3×2x−2y=12x4+(−2)y3+1=12x2y4\begin{aligned} 6x^4y^3\times2x^{-2}y &=12x^{4+(-2)}y^{3+1}\\ &=12x^2y^4 \end{aligned}

Answer. 12x2y412x^2y^4.

Check. At x=2,y=1x=2,y=1, the original gives 6(16)×2(14)=486(16)\times2(\frac14)=48, and the result gives 12(4)=4812(4)=48. Counting factors also gives x4/x2=x2x^4/x^2=x^2.

Change only multiplication to division. Subtract the complete index of each divisor, including any negative sign.

Worked example 2: dividing by a negative power

Question. Simplify 6x4y3÷(2x−2y)6x^4y^3\div(2x^{-2}y), for non-zero x,yx,y.

Plan. Divide the coefficients and subtract the divisor's indices.

6x4y3÷(2x−2y)=3x4−(−2)y3−1=3x6y2\begin{aligned} 6x^4y^3\div(2x^{-2}y) &=3x^{4-(-2)}y^{3-1}\\ &=3x^6y^2 \end{aligned}

Answer. 3x6y23x^6y^2.

Check. Multiply by the original divisor: 3x6y2×2x−2y=6x4y33x^6y^2\times2x^{-2}y=6x^4y^3. At x=2,y=1x=2,y=1, 96÷12=192=3(64)96\div\frac12=192=3(64).

If matching powers cancel completely, their quotient is 1. For example, 5a3a3=5\frac{5a^3}{a^3}=5, for a≠0a\ne0. Addition is different: a3+a3=2a3a^3+a^3=2a^3, because you collect like terms instead of joining factors.

Watch out

a4÷a−2=a4−(−2)=a6a^4\div a^{-2}=a^{4-(-2)}=a^6, not a2a^2. Also, do not combine different bases: x3y2x^3y^2 is not (xy)5(xy)^5.

Keep the coefficient calculation separate from the two index calculations.

Powers of products

A power outside a bracket applies to the complete bracket. For example, (3x2)2=(3x2)(3x2)=9x4(3x^2)^2=(3x^2)(3x^2)=9x^4: the coefficient is squared and the index on xx is multiplied by 2.

Powers of powers and products

For integer indices, where all expressions are defined,

(am)n=amn,(ab)n=anbn.(a^m)^n=a^{mn},\qquad (ab)^n=a^nb^n.

For negative integer powers, the base must be non-zero. A power of a quotient applies to its whole numerator and denominator too.

Not on the supplied formula lists: learn these laws.

For a negative coefficient, the outer power decides its sign. Squaring two negative factors gives a positive coefficient; cubing three leaves a negative one. A minus in an index instead indicates a reciprocal.

Worked example 3: squaring every factor

Question. Simplify (−2a3b−2)2(-2a^3b^{-2})^2, for non-zero a,ba,b, giving positive indices.

Plan. Square the coefficient, multiply each index by 2, then write the reciprocal factor below the line.

(−2a3b−2)2=(−2)2a3×2b−2×2=4a6b−4=4a6b4\begin{aligned} (-2a^3b^{-2})^2 &=(-2)^2a^{3\times2}b^{-2\times2}\\ &=4a^6b^{-4}\\ &=\frac{4a^6}{b^4} \end{aligned}

Answer. 4a6b4\frac{4a^6}{b^4}.

Check. At a=1,b=2a=1,b=2, the bracket is −12-\frac12, whose square is 14\frac14. The result is 416=14\frac4{16}=\frac14.

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