E2.4 - Fractional algebraic indices (Extended)

Rational power

For $x>0$ and integers $m,n$ with $n>0$,

A rational exponent represents a root and a power. Positive bases keep all rational powers real and make the index laws valid without sign ambiguities.

Products with rational indices

For $x>0$ and rational $r,s$,

The bases must be equal before you add the indices.

Not on the supplied formula lists: learn the law.

Worked example 1: fractional powers combining to an integer

Question. Simplify $4x^{1/3}\times3x^{2/3}$, for $x>0$.

Plan. Multiply 4 by 3 and add the two fractional exponents.

Answer. $12x$.

Check. At $x=64$, the original is $4(4)\times3(16)=768$. The result gives $12(64)=768$. The cube root of $x$ and its square together contain three cube-root factors, whose product is $x$.

Worked example 2: different fractional denominators

Question. Simplify $4x^{1/3}\times3x^{1/2}$, for $x>0$.

Plan. Use denominator 6 to add the exponents.

Answer. $12x^{5/6}$.

Check. At $x=64$, the original is $4(4)\times3(8)=384$. Since $64=2^6$, the result is $12(2^6)^{5/6}=12(2^5)=384$.

Watch out

In $x^{1/3}x^{1/2}$, add the fractions to get $x^{5/6}$. Multiplying the exponents would describe $(x^{1/3})^{1/2}$, a different expression equal to $x^{1/6}$.

Quotients and nested rational powers

For $x>0$ and rational $r,s$,

For positive $A,B$, $(AB)^r=A^rB^r$. The outside power applies to coefficients as well as variable powers.

Not on the supplied formula lists: learn these laws.

Worked example 3: a quotient inside a root

Question. Simplify $\left(\frac{9x^{1/2}}{x^{3/2}}\right)^{1/2}$, for $x>0$, giving positive indices.

Plan. Subtract the inner indices, then apply the outer power to both 9 and $x^{-1}$.

Answer. $\frac3{x^{1/2}}$, or $\frac3{\sqrt x}$.

Check. At $x=4$, the inner quotient is $\frac{9(2)}8=\frac94$, whose positive square root is $\frac32$. The result is $\frac3{\sqrt4}=\frac32$.