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Index Laws and Equations with Indices

How to use index laws, interpret fractional and negative powers, and solve equations by matching bases or making a substitution.

Combine powers using the correct law

a^m a^n=a^{m+n}

\frac{a^m}{a^n}=a^{m-n}

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

In the quotient, a\ne0. Fractional powers need appropriate real-domain restrictions; positive bases avoid this issue.

For a>0:

a^{1/q}=\sqrt[q]{a}

a^{p/q}=(\sqrt[q]{a})^p

Also:

a^{-m}=\frac1{a^m}

a^0=1

The zero-power rule requires a\ne0

Worked example 1

For x>0, write \frac{x^{5/3}x^{7/6}}{x^{1/2}} as a single power of x

  1. Add the numerator indices and subtract the denominator index: \frac53+\frac76-\frac12

  2. Your turn. Calculate the resulting index.

    \frac{10+7-3}{6}=\frac73

  3. Answer: x^{7/3}

Worked example 2

For a,b>0, simplify (16a^8b^{-4})^{3/4}, using positive indices.

  1. Your turn. Find 16^{3/4} by taking the fourth root first.

    (\sqrt[4]{16})^3=2^3=8

  2. Multiply indices: 8\times\frac34=6 and -4\times\frac34=-3. Thus the expression is 8a^6b^{-3}
  3. Answer: \frac{8a^6}{b^3}

Worked example 3

Work out 125^{-2/3} exactly.

  1. The cube root of 125 is 5, so 125^{2/3}=5^2=25

  2. Your turn. Use the negative index to finish the calculation.

    125^{-2/3}=\frac1{25}

  3. Answer: \frac1{25}. A negative index means a reciprocal, not a negative value.

Rewrite both sides using the same base

For a>0 and a\ne1:

a^u=a^v\quad\Longrightarrow\quad u=v

First rewrite powers such as 9 and 27 using base 3, then compare exponents.

Worked example 4

Solve 9^{x+1}=27^{2x-1}

  1. Use 9=3^2 and 27=3^3: 3^{2x+2}=3^{6x-3}

  2. Your turn. Equate exponents: 2x+2=6x-3. Find x

    4x=5, so x=\frac54

  3. Answer: x=\frac54. Both sides then equal 3^{9/2}

Worked example 5

Solve x^{-1/2}=\frac15, where x>0

  1. Rewrite as \frac1{\sqrt x}=\frac15. The positive square root must therefore equal 5

  2. Your turn. Square both sides to find x

    x=25

  3. Answer: x=25. Check 25^{-1/2}=\frac1{\sqrt{25}}=\frac15

Recognise a quadratic in a power or root

Substitution can reveal a quadratic. If u=\sqrt x, then x=u^2 and u\ge0. If u=2^x, then 2^{2x}=u^2 and u>0. Solve for u, reject impossible values, then convert back to x

Worked example 6

Solve x-6\sqrt x+8=0

  1. Let u=\sqrt x, with u\ge0. Then u^2-6u+8=0, or (u-2)(u-4)=0

  2. Your turn. Find the two possible values of u

    Give every value, separated by commas

    u=2 or u=4, both allowed.

  3. Since x=u^2, square each value.
  4. Answer: x=4 or x=16. Both satisfy the original equation.

Worked example 7

Solve 2^{2x}+2^x-6=0

  1. Let u=2^x>0. Then u^2+u-6=(u+3)(u-2)=0

  2. Your turn. Which candidate, -3 or 2, can equal 2^x?

    Keep u=2; an exponential with positive base cannot have output -3

  3. Now 2^x=2
  4. Answer: x=1. Check 4+2-6=0

Common mistakes

  • Adding indices in a power of a power. Multiply them: (x^a)^b=x^{ab}.
  • Treating a negative index as a negative answer. A negative index means take the reciprocal.
  • Raising only part of a product. Apply the power to every factor, including numerical coefficients.
  • Comparing exponents with different bases. Use the same positive base, other than 1, before equating exponents.
  • Accepting impossible substituted values. Check the substitution’s restrictions before solving for the original variable.

Now try it: Index Laws and Equations with Indices practice questions

More on this topic: Index Laws and Equations with Indices worksheet with full solutions