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Indices

Indices

Negative, zero and fractional indices

Properties Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Negative indices
  4. Fractional indices: roots Extended
  5. Simplifying algebra with these powers
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 1.7 and 2.4: negative, zero and fractional indices
  • 1.7Understand and use negative indices: \(a^{-n}=\dfrac{1}{a^n}\).
  • 1.7Understand and use fractional indices: \(a^{\frac1n}=\sqrt[n]{a}\) and \(a^{\frac mn}=\left(\sqrt[n]{a}\right)^m\). Extended
  • 1.7Evaluate numbers like \(7^{-2}\), \(81^{\frac12}\), \(8^{-\frac23}\) and \(\left(\tfrac{27}{125}\right)^{-\frac23}\) without a calculator.
  • 2.4Simplify algebra with negative and fractional powers, and write terms in the form \(ax^n\).

Before you start: the rules of indices. In 0580, Core needs negative and zero indices; fractional indices are Extended.

Formula bank

Negative indexLearn it\(a^{-n}=\dfrac1{a^n}\), \(\left(\dfrac ab\right)^{-n}=\left(\dfrac ba\right)^{n}\)means one over (flip); never a negative answer.Use it when a minus sign in the index.
Unit fraction indexLearn it\(a^{\frac1n}=\sqrt[n]a\)denominator = root.Use it when \(a^{\frac12}\), \(a^{\frac13}\).
Fraction indexLearn it\(a^{\frac mn}=\left(\sqrt[n]a\right)^{m}\)root first, then power.Use it when \(16^{\frac34}\).
Moving across the lineLearn it\(\dfrac{k}{x^{n}}=kx^{-n}\)only the x part moves; its index changes sign.Use it when "write in the form \(ax^n\)".

Negative indices

Go down the powers of 2 and each step divides by 2: \(2^2=4\), \(2^1=2\), \(2^0=1\), so \(2^{-1}=\tfrac12\), \(2^{-2}=\tfrac14\), \(2^{-3}=\tfrac18\). A negative index means one over: it makes the number small, never negative.

Power\(2^3\)\(2^2\)\(2^1\)\(2^0\)\(2^{-1}\)\(2^{-2}\)\(2^{-3}\)
Value8421\(\tfrac12\)\(\tfrac14\)\(\tfrac18\)

General methodHow to evaluate a negative power

  1. Flip the base (\(5\to\tfrac15\), \(\tfrac13\to3\)).
  2. Apply the power, now positive.
  3. Several powers of the same base? Combine the indices with the rules FIRST, then evaluate.

Example Evaluate \(3^0-3^{-1}\).

\(3^0=1\), \(3^{-1}=\tfrac13\).

  1. \(1-\tfrac13\)

Answer \(\tfrac23\)

Try it yourself

1
Value of \(5^{-2}\)?
Show answer

Answer \(\tfrac1{25}\)

2
Value of \(\left(\tfrac13\right)^{-1}\)?
Show answer

Answer 3

3
Value of \(2^{-3}\times2^4\)?
Show answer

Answer 2

Fractional indices: roots Extended

\(9^{\frac12}\times9^{\frac12}=9^1=9\), so \(9^{\frac12}\) is the square root, 3. Likewise \(8^{\frac13}=\sqrt[3]8=2\). In \(a^{\frac mn}\) the denominator is the root and the numerator is the power.

General methodHow to evaluate any power like \(a^{-\frac mn}\)

  1. Decimal base? Make it a fraction first (0.125 = \(\tfrac18\), 0.04 = \(\tfrac1{25}\)).
  2. Minus sign: FLIP the base (\(27\to\tfrac1{27}\); \(\tfrac{27}{125}\to\tfrac{125}{27}\)).
  3. Denominator: take that ROOT (of top and bottom).
  4. Numerator: raise to that POWER.

Example Find \(\left(\tfrac{27}{125}\right)^{-\frac23}\).

Flip, cube root, square.

  1. flip: \(\tfrac{125}{27}\)
  2. cube root: \(\tfrac53\)
  3. square: \(\tfrac{25}9\)

Answer \(\tfrac{25}{9}\)

Try it yourself

1
Value of \(16^{\frac34}\)?
Show answer

Answer 8

2
Value of \(8^{-\frac23}\)?
Show answer

Answer \(\tfrac14\)

3
Value of \(0.125^{-\frac23}\)?
Show answer

Answer 4

Simplifying algebra with these powers

The rules of indices work the same with negative and fractional indices - you just need careful fraction arithmetic (\(5-(-2)=7\), \(\tfrac52+\tfrac12=3\)).

General methodHow to write a term in the form \(ax^n\)

  1. Roots as powers: \(\sqrt x=x^{\frac12}\), \(\sqrt[3]{x^2}=x^{\frac23}\).
  2. Combine the x's on top (add indices) and below (add indices).
  3. Bring the bottom up: subtract its index (\(\tfrac{k}{x^{n}}=kx^{-n}\)); the number stays a fraction.
Written asIndex formWritten asIndex form
\(\sqrt{x}\)\(x^{\frac12}\)\(\dfrac1x\)\(x^{-1}\)
\(\sqrt[3]{x}\)\(x^{\frac13}\)\(\dfrac{1}{\sqrt x}\)\(x^{-\frac12}\)
\(x\sqrt{x}\)\(x^{\frac32}\)\(\dfrac{5}{x^2}\)\(5x^{-2}\)
\(\sqrt[3]{x^2}\)\(x^{\frac23}\)\(\dfrac{1}{4x^3}\)\(\tfrac14x^{-3}\)

Example Simplify \(\left(\dfrac{16x^8}{81}\right)^{\frac34}\).

Power \(\tfrac34\) on every part.

  1. \(16^{\frac34}=8\), \(81^{\frac34}=27\)
  2. \((x^8)^{\frac34}=x^6\)

Answer \(\dfrac{8x^6}{27}\)

Try it yourself

1
Simplify \(12a^5\div3a^{-2}\).
Show answer

Answer \(4a^7\)

2
Write \(\dfrac{6}{\sqrt x}\) in the form \(ax^n\).
Show answer

Answer \(6x^{-\frac12}\)

3
Simplify \(\dfrac{10x^{\frac52}}{2x^{-\frac12}}\).
Show answer

Answer \(5x^3\)

Ways they test it

Type 1 Evaluate

"Find the value of \(4^{-2}\), \(100^{-\frac12}\), \(64^{\frac23}\)." (1-2 marks each)

Spot it a number to a negative or fraction power.

  1. Flip, root, power.
  2. Exact answers only.

For example \(64^{\frac23}\)

  1. \(\sqrt[3]{64}=4\)
  2. \(4^2\)

Answer 16

Type 2 Fraction base

"Work out \(\left(\tfrac{27}{125}\right)^{-\frac23}\)." (2 marks)

Spot it a fraction in the bracket.

  1. Flip the fraction first.

Type 3 Simplify a term

"Simplify \((243x^{10})^{\frac25}\)." (2 marks)

Spot it a bracket to a fraction power.

  1. Number: root then power.
  2. Index: multiply.

For example \((243x^{10})^{\frac25}\)

  1. \(\sqrt[5]{243}=3\), \(3^2=9\)
  2. \(10\times\tfrac25=4\)

Answer \(9x^4\)

Type 4 Form \(ax^n\)

"Write \(\dfrac{4x^2\sqrt x}{2x^5}\) in the form \(ax^n\)." (3 marks)

Spot it "in the form".

  1. Roots as powers, combine, bring up.

Mistakes to avoid

  • ✗ \(5^{-2}=-25\)✓ \(\tfrac1{25}\)Negative index = one over.
  • ✗ \(16^{\frac12}=8\)✓ \(4\)Power ½ is a square root, not halving.
  • ✗ \(27^{\frac23}=18\)✓ \(9\)Cube root (3), then square.
  • ✗ \(\left(\tfrac49\right)^{-\frac12}=\tfrac23\)✓ \(\tfrac32\)Flip the fraction.
  • ✗ \(\dfrac{1}{2x^3}=2x^{-3}\)✓ \(\tfrac12x^{-3}\)Only \(x^3\) moves up.
  • ✗ \(a^5\div a^{-2}=a^3\)✓ \(a^7\)Subtracting a negative adds.

Summary

  • \(a^{-n}=\dfrac1{a^n}\): negative = flip, never a negative answer.
  • \(a^{\frac mn}\): denominator = root, numerator = power; root first.
  • Decimals → fractions before you start.
  • \(\dfrac{k}{x^n}=kx^{-n}\): only the x part moves.

If you see …

  • a minus in the indexflip the base.
  • a fraction in the indexroot, then power.

Practice

Practice: easy, medium, hard

Easy

1
Value of \(4^{-2}\)?
Show answer

Answer \(\tfrac1{16}\)

2
Value of \(36^{\frac12}\)?
Show answer

Answer 6

3
Value of \(100^{-\frac12}\)?
Show answer

Answer \(\tfrac1{10}\)

Medium

1
Value of \(64^{\frac23}\)?
Guide me

What is it asking? a value

What do we know? denominator 3 → cube root

Plan then square

Show working
  1. \(\sqrt[3]{64}=4\)
  2. \(4^2=16\)

Answer 16

2
Value of \(\left(\tfrac14\right)^{-\frac32}\)?
Guide me

What is it asking? a value

What do we know? flip first

Plan \(4^{\frac32}\)

Show working
  1. \((\sqrt4)^3=2^3\)

Answer 8

3
Value of \(2^{-2}+4^{-1}\)?
Guide me

What is it asking? a value

What do we know? no rule for adding

Plan evaluate each first

Show working
  1. \(\tfrac14+\tfrac14\)

Answer \(\tfrac12\)

Hard

1
Value of \(\left(\tfrac1{1000}\right)^{-\frac23}\)?
Guide me

What is it asking? a value

What do we know? flip, cube root, square

Plan \(1000^{\frac23}\)

Show working
  1. \(\sqrt[3]{1000}=10\)
  2. \(10^2\)

Answer 100

PartAnswer / workingMarksGuidance
\(1000^{\frac23}\) or \(10^2\)M1
100A1
Total: 2 marks
2
Simplify \((243x^{10})^{\frac25}\).
Guide me

What is it asking? one term

What do we know? power on both parts

Plan \(\sqrt[5]{243}=3\)

Show working
  1. \(3^2=9\)
  2. \(10\times\tfrac25=4\)

Answer \(9x^4\)

PartAnswer / workingMarksGuidance
\(9x^4\)B2
Total: 2 marks
3
\(y=\dfrac{4x^2\sqrt{x}}{2x^5}\). Write \(y\) in the form \(ax^n\).
Guide me

What is it asking? ax^n

What do we know? \(x^2\sqrt x=x^{\frac52}\)

Plan subtract 5

Show working
  1. \(\tfrac42=2\)
  2. \(\tfrac52-5=-\tfrac52\)

Answer \(2x^{-\frac52}\)

PartAnswer / workingMarksGuidance
\(x^{\frac52}\)M1
subtracts 5M1
\(2x^{-\frac52}\)A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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