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Indices

Indices

Indices: word problems, tough and challenge questions

Equation Solving Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. Comparing and ordering powers
  4. Adding equal powers
  5. Prime factors in index form
  6. Word problems
  7. Substitution and simultaneous index equations
  8. Ways they test it
  9. Mistakes to avoid
  10. Summary
  11. Practice
  12. Can I do this?
  13. Quiz

In this note

Syllabus 1.7 and 2.4: indices in problems
  • 1.7Compare and order powers without working them out (\(2^{30}\) or \(3^{20}\)?).
  • 1.7Simplify sums of equal powers (\(2^n+2^n=2^{n+1}\)) and use prime factors in index form.
  • 2.4Model doubling, tripling and halving with powers and solve for the number of steps.
  • 2.4Use substitution (given 2ᵃ = 3, find 8ᵃ) and solve simultaneous index equations.

You need the earlier pages, especially the unknown in the power.

Formula bank

Comparing powersLearn itmake the bases OR the indices the same\(2^{30}=8^{10}\), \(3^{20}=9^{10}\).Use it when "which is bigger" without a calculator.
Copies of one powerLearn it\(2^n+2^n=2^{n+1}\), \(3^n+3^n+3^n=3^{n+1}\)Use it when the same power added several times.
Repeated multiplierLearn it\(\text{start}\times(\text{multiplier})^{n}\)doubling × 2, tripling × 3, halving × ½.Use it when "doubles every hour".
SubstitutionLearn it\(8^a=(2^a)^3\), \(2^{a+2}=2^a\times4\)Use it when "given that 2ᵃ = 3".

Comparing and ordering powers

Same base (bigger than 1): bigger index = bigger number. Same positive index: bigger base = bigger number. If neither matches, rewrite them so one does.

General methodHow to compare two powers without a calculator

  1. Look for a common factor of the two indices (30 and 20 → 10).
  2. Rewrite each as (something)^that factor: \(2^{30}=(2^3)^{10}=8^{10}\), \(3^{20}=(3^2)^{10}=9^{10}\).
  3. Same index now: the bigger base wins.
  4. Small powers in a list: just work each one out as a fraction or whole number.

Example Show that \(2^{100}\) is greater than \(5^{40}\).

Common factor of 100 and 40 is 20.

  1. \(2^{100}=32^{20}\)
  2. \(5^{40}=25^{20}\)

Answer \(32>25\), so \(2^{100}>5^{40}\)

Adding equal powers

There is no rule for adding different powers - but adding COPIES of one power is multiplication: \(2^n+2^n=2\times2^n=2^{n+1}\); \(4^5+4^5+4^5+4^5=4^6\).

\(2^{100}+2^{100}=2^{101}\), NOT \(2^{200}\) and NOT \(4^{100}\).

Try it yourself

1
\(5^7+5^7+5^7+5^7+5^7\) as a power of 5?
Show answer

Answer \(5^8\)

2
\(2^{n}+2^{n}\) as a power of 2?
Show answer

Answer \(2^{n+1}\)

Prime factors in index form

General methodHow to find a and b in 2ᵃ × 3ᵇ = N

  1. Prime factorise N (factor tree or repeated division).
  2. Write the result with powers.
  3. Match each prime's index.

Example \(2^a\times3^b=288\). Find \(a\) and \(b\).

Prime factorise 288.

  1. \(288=2^5\times3^2\)

Answer \(a=5,\ b=2\)

Squaring or cubing a number in index form doubles or triples every index: \(2160=2^4\times3^3\times5\), so \(2160^2=2^8\times3^6\times5^2\).

Try it yourself

1
\(2^a\times3^b=72\). Find \(a\).
Show answer

Answer 3

2
\(2^a\times3^b=72\). Find \(b\).
Show answer

Answer 2

Word problems

General methodHow to model a doubling / halving / tripling problem

  1. Spot the multiplier per step (× 2, × 3, × ½) and what one step is (an hour, 10 years, a bounce).
  2. Amount after n steps = start × multiplierⁿ.
  3. To find n: divide by the start, write the result as a power of the multiplier, compare indices.
  4. Convert steps to time at the end (5 steps of 10 years = 50 years).

Example A rumour starts with 4 people and the number who have heard it doubles every hour. After n hours 1024 have heard it. Find n.

Divide by 4, then match powers of 2.

  1. \(4\times2^n=1024\)
  2. \(2^n=256=2^8\)

Answer \(n=8\)

Example A 640 g sample loses half its mass every day. After how many days is 5 g left?

Halving: × ½ each day.

  1. \(640\times\left(\tfrac12\right)^n=5\)
  2. \(2^n=128=2^7\)

Answer 7 days

Try it yourself

1
One bacterium splits into two every hour. How many after 10 hours?
Show answer

Answer 1024

2
A memory card holds 2³⁵ bytes and one photo uses 2²² bytes. How many photos fit (as a number)?
Show answer

Answer 8192

Substitution and simultaneous index equations

General methodHow to use 'given that 2ᵃ = 3'

  1. Write what you need in terms of \(2^a\): \(8^a=(2^a)^3\), \(4^{-a}=(2^a)^{-2}\), \(2^{a+2}=2^a\times2^2\).
  2. Replace \(2^a\) by 3 and work it out. You never find a itself.

Example \(3^m=5\). Find \(9^m\) and \(3^{m-1}\).

\(9^m=(3^m)^2\); \(3^{m-1}=3^m\div3\).

  1. \(5^2\)
  2. \(5\div3\)

Answer 25 and \(\tfrac53\)

General methodHow to solve two index equations together

  1. Turn each into ONE power of a base on each side.
  2. Equate indices: two linear equations.
  3. Solve them simultaneously; check both.

Example \(2^x\times4^y=64\) and \(\dfrac{25^x}{5^{y}}=125\).

Base 2 for the first, base 5 for the second.

  1. \(x+2y=6\)
  2. \(2x-y=3\)
  3. double the second and add: \(5x=12\)

Answer \(x=\tfrac{12}5,\ y=\tfrac95\)

Ways they test it

Type 1 Compare

"Without a calculator, show that \(2^{100}>5^{40}\)." (2 marks)

Spot it two powers, no calculator.

  1. Common index.
  2. Compare bases.

Type 2 Find a and b

"\(2^a\times3^b=288\)." (2 marks)

Spot it a product of prime powers.

  1. Prime factorise.

Type 3 Growth story

"…doubles every hour… find n." (3 marks)

Spot it a repeated multiplier.

  1. start × multiplierⁿ.
  2. Match powers.

For example 4 people, doubling, 1024

  1. \(2^n=256\)

Answer \(n=8\)

Type 4 Substitution

"\(3^m=5\). Find \(9^m\)."

Spot it "given that".

  1. Rewrite in terms of \(3^m\).

Mistakes to avoid

  • ✗ \(2^{100}+2^{100}=2^{200}\)✓ \(2^{101}\)Two copies = double = one more power of 2.
  • ✗ \(2^{30}<3^{20}\) because 30 > 20✓ compare at the same indexBigger index only wins with the same base.
  • ✗ Answering n when the time is asked✓ convert steps to timeEach step may be 10 years.

Summary

  • Compare powers by making the bases or the indices the same.
  • Copies of one power: \(2^n+2^n=2^{n+1}\).
  • Growth or decay by a fixed multiplier: start × multiplierⁿ.
  • Given \(2^a=3\): write everything in terms of \(2^a\).

If you see …

  • "which is bigger"a common index.
  • doubling / halvinga power of 2.

Practice

Practice: easy, medium, hard

Easy

1
\(4^5+4^5+4^5+4^5\) as a power of 4?
Show answer

Answer \(4^6\)

2
\(\dfrac{10^{220}\times10^{80}}{10^{400}}=10^k\). Find k.
Show answer

Answer −100

3
Find \(n\) when \(4^n=2^{n-1}\).
Show working
  1. \(2^{2n}=2^{n-1}\)

Answer −1

Medium

1
\(2^a=3\). Find \(8^a\).
Guide me

What is it asking? a value

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

Plan replace by 3

Show working
  1. \(3^3\)

Answer 27

2
A culture of 50 cells doubles every 20 minutes. How many cells after 2 hours?
Guide me

What is it asking? cells

What do we know? 6 doublings in 2 hours

Plan \(50\times2^6\)

Show working
  1. 50 × 64

Answer 3200

3
Solve \((\sqrt2)^x=16\).
Guide me

What is it asking? x

What do we know? \(\sqrt2=2^{\frac12}\)

Plan \(2^{\frac x2}=2^4\)

Show working
  1. \(\tfrac x2=4\)

Answer 8

Hard

1
A town's population was 200 and has tripled every 10 years since. It is now 48 600. How many years ago was it 200?
Guide me

What is it asking? years

What do we know? \(200\times3^n=48\,600\)

Plan n steps of 10 years

Show working
  1. \(3^n=243=3^5\)
  2. 5 × 10

Answer 50

PartAnswer / workingMarksGuidance
\(3^n=243\)M1
n = 5M1
50A1
Total: 3 marks
2
\(2^x=8^{y+1}\) and \(9^y=3^{x-9}\). Find \(x\).
Guide me

What is it asking? x

What do we know? \(x=3y+3\), \(2y=x-9\)

Plan substitute

Show working
  1. \(2y=3y-6\Rightarrow y=6\)
  2. \(x=3(6)+3\)

Answer 21

PartAnswer / workingMarksGuidance
both linear equationsM2
x = 21, y = 6A1
Total: 3 marks
3
A ball dropped from 1500 cm rises to half its previous height after each bounce. After how many bounces does it first rise to less than 25 cm?
Guide me

What is it asking? bounces

What do we know? \(1500\times\left(\tfrac12\right)^n<25\)

Plan try 5 and 6

Show working
  1. 5: 46.875 cm
  2. 6: 23.4375 cm

Answer 6

PartAnswer / workingMarksGuidance
halving seenM1
both valuesM1
6A1
Total: 3 marks

Can I do this?

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

Quiz

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