E1.7 - Fractional indices

E1.7 - Fractional indices

An index can tell you to take a root as well as a power. You will interpret fractional and negative fractional indices, choose a sensible order for the calculation, and find exact numerical values. This lesson is Extended only.

Fractional indices as roots

What could 642/364^{2/3} mean, and why can its value be found without a calculator? First make sense of the smaller step 641/364^{1/3}.

You already know that 43=644^3=64, so the cube root of 6464 is 44. A fractional index writes that inverse operation in power notation: 641/3=643=464^{1/3}=\sqrt[3]{64}=4. The denominator tells you which root to take.

A unit fractional index

For a positive number aa and a positive integer nn,

a1/n=ana^{1/n}=\sqrt[n]{a}

The symbol means the positive root of order nn when a>0a>0. In particular, a1/2=aa^{1/2}=\sqrt a. Also, 01/n=00^{1/n}=0.

For a negative base, an odd root is real, such as (−27)1/3=−3(-27)^{1/3}=-3. An even root of a negative number has no real value.

Why does the fraction fit the index law? Multiplying the index 13\frac13 by 33 gives 11, so cubing 641/364^{1/3} must recover 6464. Taking a cube root does exactly that. The square-root symbol chooses the nonnegative root: 491/2=749^{1/2}=7, although both 77 and −7-7 square to 4949.

Worked example 1: a square root as an index

Question. Find the value of 811/281^{1/2}.

Plan. Read the denominator 22 as a square root.

811/2=81index one half means square root=9the nonnegative root\begin{aligned} 81^{1/2}&=\sqrt{81} && \text{index one half means square root}\\ &=9 && \text{the nonnegative root} \end{aligned}

Answer. 99.

Check. 92=819^2=81, and 99 is nonnegative.

Keep the base 8181, but change the denominator of the index from 22 to 44. We now need a fourth root.

Worked example 2: changing the root

Question. Find the value of 811/481^{1/4}.

Plan. Look for the positive number whose fourth power is 8181.

811/4=814the denominator gives the root=33×3×3×3=81\begin{aligned} 81^{1/4}&=\sqrt[4]{81} && \text{the denominator gives the root}\\ &=3 && 3\times3\times3\times3=81 \end{aligned}

Answer. 33.

Check. 34=813^4=81. Also, taking a square root twice gives 81=9=3\sqrt{\sqrt{81}}=\sqrt9=3.

Watch out

811/4=81÷481^{1/4}=81\div4 is wrong: an index is not a multiplier. Here it asks for a fourth root. Check by raising your answer to the fourth power.

A new base lets you test the same interpretation.

Powers of roots

An index such as 23\frac23 combines two instructions: take a cube root and square the result. The denominator names the root; the numerator names the power.

A positive fractional index

For a positive base aa and positive integers m,nm,n,

am/n=(an)m=amna^{m/n}=\left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}

These forms also give zero when a=0a=0 and m/n>0m/n>0. Root first is often easier.

Not given in the List of formulas: learn this rule.

For example, 82/3=(81/3)2=22=48^{2/3}=(8^{1/3})^2=2^2=4. Taking the root first keeps the numbers small; cubing or squaring a large base first is usually unnecessary.

For a negative base, first write the fractional index in its simplest form. A reduced odd denominator allows a real root, for example (−8)2/3=(−2)2=4(-8)^{2/3}=(-2)^2=4. A reduced even denominator does not. Use the general formula above freely for positive bases; do not rearrange roots and powers indiscriminately for negative bases.

Worked example 3: root first, then square

Question. Find the value of 272/327^{2/3}.

Plan. Take the cube root of 2727, then square it.

272/3=(273)2root from the denominator=32cube root of 27 is 3=9power from the numerator\begin{aligned} 27^{2/3}&=\left(\sqrt[3]{27}\right)^2 && \text{root from the denominator}\\ &=3^2 && \text{cube root of }27\text{ is }3\\ &=9 && \text{power from the numerator} \end{aligned}

Answer. 99.

Check. 93=7299^3=729 and 272=72927^2=729, so cubing the result agrees with squaring the base.

Keep the same base and root, but change the numerator from 22 to 44. Only the final power changes.

Worked example 4: changing the numerator

Question. Find the value of 274/327^{4/3}.

Plan. Take the same cube root, then raise it to the fourth power.

274/3=(273)4denominator still means cube root=34numerator now means fourth power=81evaluate\begin{aligned} 27^{4/3}&=\left(\sqrt[3]{27}\right)^4 && \text{denominator still means cube root}\\ &=3^4 && \text{numerator now means fourth power}\\ &=81 && \text{evaluate} \end{aligned}

Answer. 8181.

Check. The indices 43\frac43 and 23\frac23 differ by 23\frac23. The quotient 81÷9=981\div9=9 is 272/327^{2/3}, as the quotient law predicts.

Watch out

272/3=3327^{2/3}=3^3 confuses the jobs of the two numbers. The 33 in the denominator tells you to take a cube root; the 22 in the numerator tells you to square that root.

The opening question now has an answer: 642/3=42=1664^{2/3}=4^2=16. You need only recognise a cube and a square.

Negative fractional indices

A negative fractional index adds one more instruction: take the reciprocal. For example, 16−1/216^{-1/2} means the reciprocal of the square root of 1616, so its value is 14\frac14.

A negative fractional index

For a positive base aa and positive integers m,nm,n,

a−m/n=1am/n=1(an)ma^{-m/n}=\frac1{a^{m/n}} =\frac1{\left(\sqrt[n]{a}\right)^m}

The base cannot be zero: the reciprocal would divide by zero. Negative bases can be used when the fractional index in simplest form has an odd denominator, using its real odd root.

Not given in the List of formulas: learn this rule.

Read the three jobs separately: root from the denominator, power from the numerator, reciprocal from the minus sign. You can take the reciprocal first instead, but root first usually gives simpler arithmetic.

Worked example 5: a positive fractional power

Question. Find the value of 322/532^{2/5}.

Plan. Find the fifth root of 3232, then square it.

322/5=(325)2take the fifth root=22=425=32\begin{aligned} 32^{2/5}&=\left(\sqrt[5]{32}\right)^2 && \text{take the fifth root}\\ &=2^2=4 && 2^5=32 \end{aligned}

Answer. 44.

Check. 45=10244^5=1024 and 322=102432^2=1024, as the fractional index requires.

Change only the sign of the index. The root and power stay the same; their result becomes the denominator.

Worked example 6: changing the sign of the index

Question. Find the value of 32−2/532^{-2/5}.

Plan. Evaluate the positive fractional power, then take its reciprocal.

32−2/5=1322/5negative index means reciprocal=1(325)2root then power=14the denominator is 22\begin{aligned} 32^{-2/5}&=\frac1{32^{2/5}} && \text{negative index means reciprocal}\\ &=\frac1{(\sqrt[5]{32})^2} && \text{root then power}\\ &=\frac14 && \text{the denominator is }2^2 \end{aligned}

Answer. 14\frac14.

Check. The opposite powers multiply to 11: 322/5×32−2/5=4×14=132^{2/5}\times32^{-2/5}=4\times\frac14=1.

Watch out

32−2/5=−432^{-2/5}=-4 is wrong. A negative index changes a nonzero value to its reciprocal; it does not change its sign. With a positive base, all the real powers used here are positive.

Use the same three instructions with a cube root.

Quick Recap

  • The denominator of a fractional index tells you which root to take.
  • The numerator tells you the power; taking the root first often saves work.
  • A negative index means reciprocal, so the base must be nonzero.
  • Positive bases support all the fractional-index rules here. A negative base needs a reduced odd denominator for a real value.

The opening value is 642/3=1664^{2/3}=16: cube root first, then square.