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Indices

Indices

The rules of indices

Properties Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. What a power is
  4. The rules of indices
  5. Terms with numbers and several letters
  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: the rules of indices
  • 1.7Read and write powers: base, index, and what \(a^n\) means.
  • 1.7Use the rules of indices to multiply, divide and raise powers to a power, with numbers and with letters.
  • 1.7Know that anything (except 0) to the power 0 is 1.
  • 2.4Simplify algebraic terms such as \((3p^2)^5\), \(18x^2y^6\div2xy^2\) and \(\dfrac{(2a^3b)^4}{(6a^5b^3)^2}\).

Next: negative and fractional indices. Paper 1 has no calculator, so learn the powers table below.

Formula bank

Multiply: addLearn it\(a^m\times a^n=a^{m+n}\)same base only.Use it when powers multiplied.
Divide: subtractLearn it\(a^m\div a^n=a^{m-n}\)same base only.Use it when powers divided.
Power of a power: multiplyLearn it\((a^m)^n=a^{mn}\)Use it when a power outside a bracket.
Zero indexLearn it\(a^0=1\) \((a\neq0)\)Use it when a power of 0.
Power of a product / fractionLearn it\((ab)^n=a^nb^n\), \(\left(\dfrac ab\right)^n=\dfrac{a^n}{b^n}\)EVERYTHING inside gets the power.Use it when a bracket with a number and letters.

What a power is

A power is repeated multiplication: \(2^5=2\times2\times2\times2\times2=32\). The 2 is the base; the 5 is the index (power), the number of copies. \(2^5\) is NOT \(2\times5\), and \(x^3\) means \(x\times x\times x\), not \(3x\).

Learn these by heart (Paper 1 has no calculator):

BasePowers
2\(2,\ 4,\ 8,\ 16,\ 32,\ 64,\ 128,\ 256,\ 512,\ 1024\)
3\(3,\ 9,\ 27,\ 81,\ 243,\ 729\)
4\(4,\ 16,\ 64,\ 256,\ 1024\)
5\(5,\ 25,\ 125,\ 625,\ 3125\)
6, 7\(6^2=36,\ 6^3=216\); \(7^2=49,\ 7^3=343\)
10\(10^2=100,\ 10^3=1000,\ 10^6=\) one million

One number can be a power of several bases: \(64=2^6=4^3=8^2\), \(81=3^4=9^2\), \(16=2^4=4^2\).

The rules of indices

Every rule comes from counting copies of the base: \(x^3\times x^2=(xxx)(xx)=x^5\); \(\tfrac{x^5}{x^2}\) cancels two copies to leave \(x^3\); \((x^3)^2=x^3\times x^3=x^6\); and \(\tfrac{x^3}{x^3}=1=x^0\).

General methodHow to simplify any power of one base

  1. Check the bases are the same. If not, rewrite one (4 = 2², 9 = 3², 8 = 2³) or leave it.
  2. Brackets first: multiply the indices.
  3. Multiplying: add the indices. Dividing: subtract them.
  4. Power 0 → 1. Power 1 → just the base.

Example Write \(4^3\times2^5\) as a single power of 2.

Rewrite 4 as \(2^{2}\) first.

  1. \(4^3=(2^2)^3=2^6\)
  2. \(2^6\times2^5\)

Answer \(2^{11}\)

No rule for adding: \(a^2+a^3\) does not simplify, and \(42+7\) as a power of 7 is just \(49=7^{2}\).

Try it yourself

1
Simplify \(7^5\times7^6\) (as a power of 7).
Show answer

Answer \(7^{11}\)

2
Simplify \(t^{10}\times t^4\div t^3\).
Show answer

Answer \(t^{11}\)

3
\(6^2\times6^n=6^6\). Find n.
Show answer

Answer 4

Terms with numbers and several letters

General methodHow to simplify a term like 18x²y⁶ ÷ 2xy²

  1. Numbers: multiply or divide them as ordinary numbers. A bracket to a power raises the number too (3⁵, not 3 × 5).
  2. Each letter in turn: use the rules on that letter only (a letter on its own has power 1).
  3. Write one term: number first, letters in alphabetical order; a letter with power 0 disappears.

Example Simplify \((3p^2)^5\).

Both the 3 and the \(p^{2}\) get the power 5.

  1. \(3^5=243\)
  2. \((p^2)^5=p^{10}\)

Answer \(243p^{10}\)

Example Simplify \(\dfrac{(4x^3y^2)^3}{8x^5y}\).

Bracket first, then divide each part.

  1. Top: \(64x^9y^6\)
  2. \(64\div8=8\); \(x^{9-5}\); \(y^{6-1}\)

Answer \(8x^4y^5\)

Try it yourself

1
Simplify \(5x^3y\times4x^2y^4\).
Show answer

Answer \(20x^5y^5\)

2
Simplify \(18x^2y^6\div2xy^2\).
Show answer

Answer \(9xy^4\)

3
Value of \(4x^0\)?
Show answer

Answer 4

Ways they test it

Type 1 One-mark simplify

"Simplify \(p^3\times p^8\)", "\(m^{20}\div m^4\)", "\((k^4)^6\)" (1 mark each)

Spot it one letter, one operation.

  1. Add, subtract or multiply the indices.

For example \((k^4)^6\)

  1. 4 × 6

Answer \(k^{24}\)

Type 2 Term with a coefficient

"Simplify \((3p^2)^5\)" (2 marks)

Spot it a number inside a bracket.

  1. Number to the power; letters multiply.

Type 3 Simplify fully

"Simplify fully \(\dfrac{(2a^3b)^4\times3ab^2}{(6a^5b^3)^2}\)" (3 marks)

Spot it brackets top and bottom.

  1. Expand each bracket.
  2. Divide numbers and each letter.

For example \(\tfrac{48a^{13}b^6}{36a^{10}b^6}\)

    Answer \(\tfrac43a^3\)

    Type 4 Missing index

    "\(5^n\times5^3=5^{11}\). Find n."

    Spot it an unknown power.

    1. Same base both sides: compare indices.

    Type 5 Single power

    "Write \(2^3\times8^2\) as a power of 2."

    Spot it different bases.

    1. Rewrite the bases first.

    Mistakes to avoid

    • ✗ \(3^2\times3^4=9^6\)✓ \(3^6\)Keep the base; add the indices.
    • ✗ \((x^3)^2=x^5\)✓ \(x^6\)A power of a power multiplies.
    • ✗ \(x^6\div x^2=x^3\)✓ \(x^4\)Subtract, don't divide.
    • ✗ \((3p^2)^5=15p^{10}\)✓ \(243p^{10}\)The 3 is raised to the power 5.
    • ✗ \(2x^0=1\)✓ \(2\)Only the \(x\) has the power 0.

    Summary

    • Multiply → add; divide → subtract; bracket → multiply.
    • \(a^0=1\); a bracket's power goes on everything inside.
    • Same base only: rewrite 4, 8, 9, 27 … as powers of 2 or 3.
    • Numbers and letters separately; number first.

    If you see …

    • "as a single power of 2"rewrite every base as a power of 2.
    • a sum of powerswork it out as a number.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Simplify \(h^2\times h^9\).
    Show answer

    Answer \(h^{11}\)

    2
    Simplify \(m^{20}\div m^4\).
    Show answer

    Answer \(m^{16}\)

    3
    Value of \(9^0+9^1\)?
    Show answer

    Answer 10

    Medium

    1
    Simplify \((5y^4)^3\).
    Guide me

    What is it asking? one term

    What do we know? bracket to the power 3

    Plan \(5^3\), \(y^{12}\)

    Show working
    1. \(5^3=125\)
    2. \(4\times3=12\)

    Answer \(125y^{12}\)

    2
    Simplify \(\dfrac{24x^8}{6x^2}\).
    Guide me

    What is it asking? one term

    What do we know? divide numbers, subtract indices

    Plan 24 ÷ 6, 8 − 2

    Show working
    1. \(4x^{6}\)

    Answer \(4x^6\)

    3
    \(2^3\times8^2=2^k\). Find k.
    Guide me

    What is it asking? k

    What do we know? \(8=2^3\)

    Plan \(8^2=2^6\)

    Show working
    1. \(2^3\times2^6=2^9\)

    Answer 9

    Hard

    1
    Simplify \(\dfrac{(3m^2n)^3}{9mn^3}\).
    Guide me

    What is it asking? one term

    What do we know? bracket first

    Plan \(27m^6n^3\)

    Show working
    1. \(27m^6n^3\div9mn^3\)

    Answer \(3m^5\)

    PartAnswer / workingMarksGuidance
    \(27m^6n^3\) seenB1
    \(3m^5\)B1
    Total: 2 marks
    2
    Simplify \((5x^3)^2\times2x^4\).
    Guide me

    What is it asking? one term

    What do we know? bracket, then multiply

    Plan \(25x^6\times2x^4\)

    Show working
    1. \(25x^6\)
    2. \(50x^{10}\)

    Answer \(50x^{10}\)

    PartAnswer / workingMarksGuidance
    \(50x^{10}\)B2
    Total: 2 marks
    3
    Simplify fully \(\dfrac{(2a^3b)^4\times3ab^2}{(6a^5b^3)^2}\).
    Guide me

    What is it asking? one term

    What do we know? expand both brackets

    Plan top \(48a^{13}b^6\), bottom \(36a^{10}b^6\)

    Show working
    1. \(16a^{12}b^4\times3ab^2=48a^{13}b^6\)
    2. \(36a^{10}b^6\)
    3. \(\tfrac{48}{36}a^3b^0\)

    Answer \(\tfrac43a^3\)

    PartAnswer / workingMarksGuidance
    topM1
    bottomM1
    \(\tfrac43a^3\)A1
    Total: 3 marks

    Can I do this?

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

    Quiz

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