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Maths · Number

Zero, negative and fractional indices

What could 2 to the power 0, a negative power or a fractional power possibly mean? Following the index laws backwards gives every one of them a value - a zero index gives 1, a negative index gives a reciprocal, and (at Higher) a fractional index gives a root.

  • 5 key terms
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Last Lesson

Answer each one, then check.

  1. 1

    Write \(2^5 \times 2^3\) as a single power of 2.

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    \(2^8\)

  2. 2

    Write \(3^7 \div 3^2\) as a single power of 3.

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    \(3^5\)

  3. 3

    Write \((5^2)^3\) as a single power of 5.

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    \(5^6\)

  4. 4

    What is \(\sqrt[3]{27}\)?

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    \(3\)

  5. 5

    What is the reciprocal of 4?

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    \(\frac{1}{4}\) - one divided by the number.

Learning Objectives

  1. 1Know that any number (except 0) to the power 0 is 1.
  2. 2Work out negative powers as reciprocals.
  3. 3Work out negative powers of fractions.
  4. 4Work out fractional powers as roots. (Higher)
  5. 5Work out powers such as \(27^{-\frac{2}{3}}\). (Higher)

Follow the Pattern

Each step down, the index goes down by 1 and the value is divided by 2.

  • \(2^3\)

    Value: \(8\)

  • \(2^2\)

    Value: \(4\). How it continues: \(8 \div 2\)

  • \(2^1\)

    Value: \(2\). How it continues: \(4 \div 2\)

  • \(2^0\)

    Value: \(1\). How it continues: \(2 \div 2\)

  • \(2^{-1}\)

    Value: \(\frac{1}{2}\). How it continues: \(1 \div 2\)

  • \(2^{-2}\)

    Value: \(\frac{1}{4}\). How it continues: \(\frac{1}{2} \div 2\)

  • \(2^{-3}\)

    Value: \(\frac{1}{8}\). How it continues: \(\frac{1}{4} \div 2\)

The Zero Index

Any number except 0 raised to the power 0 is 1.

  • Using the laws

    \(5^3 \div 5^3 = 5^{3-3} = 5^0\). But anything divided by itself is 1, so \(5^0 = 1\).

  • Examples

    \(7^0 = 1\), \(1000^0 = 1\), \((-3)^0 = 1\), \(\left(\frac{2}{3}\right)^0 = 1\).

  • Careful

    \(3 \times 4^0 = 3 \times 1 = 3\), not 1.

Negative Indices

A negative index means "one over": \(a^{-n} = \dfrac{1}{a^n}\).

  • Whole numbers

    \(4^{-2} = \dfrac{1}{4^2} = \dfrac{1}{16}\). \(10^{-3} = \dfrac{1}{1000} = 0.001\).

  • It is not negative

    \(2^{-3} = \frac{1}{8}\), a small positive number, not \(-8\).

  • Fractions flip

    \(\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}\). A negative power turns a fraction upside down.

  • Using the laws

    \(2^3 \div 2^5 = 2^{-2}\), and writing it out: \(\dfrac{2 \times 2 \times 2}{2 \times 2 \times 2 \times 2 \times 2} = \dfrac{1}{4}\).

Negative Powers

Work out (a) \(4^{-3}\) (b) \(\left(\frac{3}{5}\right)^{-2}\)

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  1. 1 (a) Negative power: one over \(4^{-3} = \dfrac{1}{4^3}\)
  2. 2 Work out the power \(4^3 = 64\), so \(4^{-3} = \dfrac{1}{64}\)
  3. 3 (b) Negative power of a fraction: flip it \(\left(\dfrac{3}{5}\right)^{-2} = \left(\dfrac{5}{3}\right)^2\)
  4. 4 Square the top and the bottom \(\dfrac{5^2}{3^2} = \dfrac{25}{9}\)

Answer(a) \(\dfrac{1}{64}\) (b) \(\dfrac{25}{9} = 2\dfrac{7}{9}\)

Powers That Are Roots

A power of one-half is a square root; a power of one-third is a cube root.

  • Why

    \(9^{\frac{1}{2}} \times 9^{\frac{1}{2}} = 9^1 = 9\). The number that multiplies by itself to give 9 is 3, so \(9^{\frac{1}{2}} = \sqrt{9} = 3\).

  • Unit fractions

    \(a^{\frac{1}{2}} = \sqrt{a}\), \(a^{\frac{1}{3}} = \sqrt[3]{a}\), and \(a^{\frac{1}{n}} = \sqrt[n]{a}\).

  • Examples

    \(25^{\frac{1}{2}} = 5\), \(64^{\frac{1}{3}} = 4\), \(81^{\frac{1}{4}} = 3\), \(32^{\frac{1}{5}} = 2\).

  • Other fractions

    \(a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m\): the bottom is the root, the top is the power. \(8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4\).

Three Steps for a Negative Fractional Power

Work out \(27^{-\frac{2}{3}}\) one part of the index at a time.

  1. 1 Minus: flip

    \(27^{-\frac{2}{3}} = \dfrac{1}{27^{\frac{2}{3}}}\)
  2. 2 Bottom: root

    The 3 means cube root: \(\sqrt[3]{27} = 3\)
  3. 3 Top: power

    The 2 means square: \(3^2 = 9\)
  4. 4 Answer

    \(27^{-\frac{2}{3}} = \dfrac{1}{9}\)

A Fraction to a Fractional Power

Work out \(\left(\dfrac{16}{81}\right)^{-\frac{3}{4}}\)

Show the solutionHide the solution
  1. 1 Minus: flip the fraction \(\left(\dfrac{81}{16}\right)^{\frac{3}{4}}\)
  2. 2 Bottom of the index, 4: fourth root of the top and the bottom \(\sqrt[4]{81} = 3\), \(\sqrt[4]{16} = 2\)
  3. 3 So far \(\left(\dfrac{3}{2}\right)^3\)
  4. 4 Top of the index, 3: cube \(\dfrac{3^3}{2^3} = \dfrac{27}{8}\)

Answer\(\dfrac{27}{8} = 3\dfrac{3}{8}\)

Solving with Powers

Find \(x\) if \(8^x = \dfrac{1}{4}\)

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  1. 1 Write both sides as powers of 2 \(8 = 2^3\) and \(\dfrac{1}{4} = 2^{-2}\)
  2. 2 So \((2^3)^x = 2^{-2}\), that is \(2^{3x} = 2^{-2}\)
  3. 3 The bases match, so the indices are equal \(3x = -2\)
  4. 4 Divide by 3 \(x = -\dfrac{2}{3}\)

Answer\(x = -\dfrac{2}{3}\)

Match the Power to Its Value

  • \(5^0\)

    \(1\)

  • \(2^{-3}\)

    \(\frac{1}{8}\)

  • \(49^{\frac{1}{2}}\)

    \(7\)

  • \(8^{\frac{2}{3}}\)

    \(4\)

  • \(1000^{\frac{1}{3}}\)

    \(10\)

  • \(\left(\frac{1}{3}\right)^{-2}\)

    \(9\)

Case study

Carbon Dating and Negative Powers

Living things take in a little radioactive carbon-14 while they are alive. After they die it decays, and every 5730 years - its half-life - half of what is left disappears. After 1 half-life, \(2^{-1} = \frac{1}{2}\) of the carbon-14 remains; after 2, \(2^{-2} = \frac{1}{4}\); after \(n\) half-lives, \(2^{-n}\). By measuring how much is left, scientists can work out how long ago something died. A sample with \(\frac{1}{8} = 2^{-3}\) of its carbon-14 left is about \(3 \times 5730 = 17\,190\) years old.

5730 Years: the half-life of carbon-14
\(2^{-3} = \frac{1}{8}\) What is left after 3 half-lives (about 17 190 years)

Index Sort

Sort these into three groups - "less than 1", "equal to 1" and "more than 1" - then work out each value. \(3^0\), \(5^{-1}\), \(\left(\frac{1}{2}\right)^{-2}\), \(10^{-2}\), \(\left(\frac{3}{4}\right)^0\), \(\left(\frac{2}{5}\right)^{-1}\). Higher: add \(16^{\frac{1}{2}}\), \(8^{-\frac{1}{3}}\), \(27^{\frac{2}{3}}\) and \(\left(\frac{1}{4}\right)^{-\frac{1}{2}}\).

1. Predict the group first.

2. Then work out the value.

3. Explain one surprise.

A good answer shows: Less than 1: \(5^{-1} = \frac{1}{5}\), \(10^{-2} = \frac{1}{100}\), and (Higher) \(8^{-\frac{1}{3}} = \frac{1}{2}\). Equal to 1: \(3^0\), \(\left(\frac{3}{4}\right)^0\). More than 1: \(\left(\frac{1}{2}\right)^{-2} = 4\), \(\left(\frac{2}{5}\right)^{-1} = \frac{5}{2}\), and (Higher) \(16^{\frac{1}{2}} = 4\), \(27^{\frac{2}{3}} = 9\), \(\left(\frac{1}{4}\right)^{-\frac{1}{2}} = 2\).

Can I...?

  1. 1Explain why \(a^0 = 1\).
  2. 2Work out a negative power of a whole number.
  3. 3Work out a negative power of a fraction.
  4. 4Write \(\frac{1}{a^n}\) as a negative power.
  5. 5Work out \(a^{\frac{1}{n}}\) as a root. (Higher)
  6. 6Work out \(a^{\frac{m}{n}}\). (Higher)
  7. 7Work out negative fractional powers. (Higher)
  8. 8Solve equations such as \(8^x = \frac{1}{4}\). (Higher)

Summary & Exam Focus

  • \(a^0 = 1\) for any \(a\) except 0.
  • \(a^{-n} = \dfrac{1}{a^n}\); a negative power of a fraction flips it.
  • (Higher) \(a^{\frac{1}{n}} = \sqrt[n]{a}\).
  • (Higher) \(a^{\frac{m}{n}}\): root with the bottom, power with the top; a minus sign flips.

Exam focus

Find the value of \(125^{-\frac{2}{3}}\) (2 marks) (2 marks)

Do the three parts in this order - minus (flip), bottom (root), top (power) - and write each step. Taking the root before the power keeps the numbers small.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Zero index
Any non-zero number to the power 0 equals 1.
Negative index
\(a^{-n} = \dfrac{1}{a^n}\): one over the positive power.
Reciprocal
One divided by a number. The reciprocal of 4 is \(\frac{1}{4}\); of \(\frac{2}{3}\) is \(\frac{3}{2}\).
Fractional index (Higher)
A power that is a fraction: the bottom is a root, the top is a power.
nth root
The number that, raised to the power \(n\), gives the original number.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 1 mark

    Write down the value of \(7^0\).

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    Model answer

    \(1\)

    Mark scheme

    • \(1\) — B1
  2. Question 2 Non-calculator 1 mark

    Write \(4^{-2}\) as a fraction.

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    Model answer

    \(\dfrac{1}{16}\)

    Mark scheme

    • \(\frac{1}{16}\) — B1
  3. Question 3 Non-calculator · Higher 1 mark

    Write down the value of \(64^{\frac{1}{2}}\).

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    Model answer

    \(8\)

    Mark scheme

    • \(8\) — B1
  4. Question 4 Non-calculator · Higher 2 marks

    Find the value of \(125^{-\frac{2}{3}}\).

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    Model answer

    \(125^{-\frac{2}{3}} = \dfrac{1}{125^{\frac{2}{3}}}\). \(\sqrt[3]{125} = 5\) and \(5^2 = 25\), so the answer is \(\dfrac{1}{25}\).

    Mark scheme

    • \(\dfrac{1}{125^{\frac{2}{3}}}\), or \(\sqrt[3]{125} = 5\), or 25 seen — M1
    • \(\frac{1}{25}\) — A1
  5. Question 5 Non-calculator · Higher 2 marks

    Find the value of \(\left(\dfrac{27}{8}\right)^{-\frac{2}{3}}\).

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    Model answer

    Flip: \(\left(\dfrac{8}{27}\right)^{\frac{2}{3}}\). Cube root: \(\dfrac{2}{3}\). Square: \(\dfrac{4}{9}\).

    Mark scheme

    • \(\left(\frac{8}{27}\right)^{\frac{2}{3}}\), or \(\frac{3}{2}\) or \(\frac{2}{3}\) seen — M1
    • \(\frac{4}{9}\) — A1
  6. Question 6 Non-calculator · Higher 3 marks

    Write \(\sqrt{8} \times 4^{-3}\) as a single power of 2.

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    Model answer

    \(\sqrt{8} = 8^{\frac{1}{2}} = (2^3)^{\frac{1}{2}} = 2^{\frac{3}{2}}\). \(4^{-3} = (2^2)^{-3} = 2^{-6}\). \(2^{\frac{3}{2}} \times 2^{-6} = 2^{\frac{3}{2} - 6} = 2^{-\frac{9}{2}}\).

    Mark scheme

    • \(\sqrt{8}\) written as \(2^{\frac{3}{2}}\) — M1
    • \(4^{-3}\) written as \(2^{-6}\) — M1
    • \(2^{-\frac{9}{2}}\) — A1

Quick check

  1. What is the value of \(2^{-2}\)?

    1. A\(-4\)
    2. B\(\frac{1}{4}\)
    3. C\(-\frac{1}{4}\)
    4. D\(4\)
    Show answerHide answer

    B: \(\frac{1}{4}\)

    A negative power means one over: \(2^{-2} = \dfrac{1}{2^2} = \dfrac{1}{4}\).

  2. What is the value of \(6^0\)?

    1. A\(1\)
    2. B\(0\)
    3. C\(6\)
    4. D\(60\)
    Show answerHide answer

    A: \(1\)

    Any number (except 0) to the power 0 is 1.

  3. (Higher) What is the value of \(8^{\frac{2}{3}}\)?

    1. A\(\frac{16}{3}\)
    2. B\(64\)
    3. C\(2\)
    4. D\(4\)
    Show answerHide answer

    D: \(4\)

    Cube root first: \(\sqrt[3]{8} = 2\). Then square: \(2^2 = 4\).

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