Indices
Solving equations: the unknown in the base
In this note
- 2.4Solve equations where the power is known and the base is not: \(x^3=64\), \(x^{-2}=\tfrac19\), \(x^{\frac32}=8\).
- 2.4Undo any power by raising both sides to the reciprocal power; know when there are two answers.
- 2.4Find unknown numbers in identities such as \((ax^n)^3=8x^{15}\) and write roots as powers (\(\sqrt{x^3}=x^n\)).
Compare with the unknown in the power. Ask first: is the letter UP in the index, or DOWN in the base?
Formula bank
Which type is it?
| Equation | Where is the unknown? | Method |
|---|---|---|
| \(2^x=32\) | in the power (up) | match the bases, compare indices |
| \(x^2=32\) | in the base (down) | undo the power: raise both sides to \(\tfrac12\) |
| \((x+1)^3=27\) | in the base | cube root both sides |
Undoing a power
Raising to the reciprocal power undoes a power: \(\left(x^{\frac mn}\right)^{\frac nm}=x^{1}=x\). Power 5 → \(\tfrac15\); power \(\tfrac32\) → \(\tfrac23\); power \(-2\) → \(-\tfrac12\).
General methodHow to solve any equation with the unknown in the base
- Isolate the power: \(2x^5=64\Rightarrow x^5=32\) (divide the number off first).
- Raise both sides to the reciprocal of the power.
- Evaluate the right side with flip, root, power.
- Even root taken? Give \(\pm\) (unless x must be positive, e.g. a length). Odd root: one answer.
- Check by substituting back.
Example Solve \(8x^{\frac32}=27\).
Isolate, then raise to \(\tfrac23\).
- \(x^{\frac32}=\tfrac{27}{8}\)
- \(x=\left(\tfrac{27}8\right)^{\frac23}=\left(\tfrac32\right)^2\)
Answer \(x=\tfrac94\)
Example Solve \(t^{-2}=9\).
Flip both sides, then square root (±).
- \(t^2=\tfrac19\)
Answer \(t=\pm\tfrac13\)
Try it yourself
Show answer
Answer −2
Show answer
Answer \(\pm7\)
Show answer
Answer 4
Show answer
Answer \(\tfrac12\)
Finding a and n, and roots as powers
General methodHow to find a and n in \((ax^n)^p=bx^q\)
- Expand: \(a^{p}x^{np}\).
- Numbers: \(a^{p}=b\) → undo the power.
- Indices: \(np=q\) → divide.
Example \((ax^n)^3=8x^{15}\). Find \(a\) and \(n\).
\(a^3x^{3n}=8x^{15}\).
- \(a^3=8\)
- \(3n=15\)
Answer \(a=2,\ n=5\)
Try it yourself
Show answer
Answer \(\tfrac32\)
Show answer
Answer \(\tfrac52\)
Ways they test it
Type 1 Simple
"Find x: \(x^3=125\); \(x^{-1}=4\)." (1 mark each)
Spot it a power of x equals a number.
- Reciprocal power.
Type 2 Number in front
"Solve \(4y^{-2}=1\)." (2 marks)
Spot it a coefficient.
- Divide first.
- Then flip and root (±).
For example \(4y^{-2}=1\)
- \(y^{-2}=\tfrac14\)
- \(y^2=4\)
Answer \(y=\pm2\)
Type 3 Fraction power
"Solve \(8x^{\frac32}=27\)." (3 marks)
Spot it a fraction in the index.
- Isolate; raise to \(\tfrac23\).
Type 4 Identity
"\((ax^n)^{\frac32}=27x^9\). Find a and n." (3 marks)
Spot it two unknowns.
- Match number and index.
Mistakes to avoid
- ✗ \(x^2=49\Rightarrow x=7\)✓ \(x=\pm7\)Even power: the negative works too.
- ✗ \(2x^5=64\Rightarrow x=\sqrt[5]{64}\)✓ \(x=2\)Isolate the power first.
- ✗ \(x^{\frac32}=8\Rightarrow x=12\)✓ \(x=4\)Raise to \(\tfrac23\); don't multiply.
- ✗ \(x^{-2}=\tfrac19\Rightarrow x=-\tfrac13\)✓ \(x=\pm3\)Flip both sides.
Summary
- Unknown in the base → reciprocal power. Unknown in the power → match bases.
- Isolate the power before anything else.
- Even root → ±; odd root → one answer.
- \((ax^n)^p\): \(a^{p}\) and \(np\).
If you see …
- a length or "x > 0"positive answer only.
- \(x^{-n}\)flip both sides first.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 5
Show answer
Answer \(\tfrac14\)
Show answer
Answer 100
Medium
Guide me
What is it asking? y
What do we know? divide by 4
Plan \(y^{2}=4\)
Show working
- \(y^{-2}=\tfrac14\)
- \(y^2=4\)
Answer \(\pm2\)
Guide me
What is it asking? x
What do we know? \(x^3=-27\)
Plan odd root: one answer
Show working
- \(\sqrt[3]{-27}\)
Answer −3
Guide me
What is it asking? x
What do we know? raise to \(\tfrac43\)
Plan \(\sqrt[3]{27}=3\), \(3^4\)
Show working
- \(27^{\frac43}=81\)
Answer 81
Hard
Guide me
What is it asking? x
What do we know? raise to \(\tfrac23\)
Plan cube root \(\tfrac23\), square
Show working
- \(\left(\tfrac8{27}\right)^{\frac23}=\left(\tfrac23\right)^2\)
Answer \(\tfrac49\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(x=\left(\tfrac8{27}\right)^{\frac23}\) | M1 | ||
| \(\tfrac49\) | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? a
What do we know? \(a^{\frac32}=27\)
Plan raise to \(\tfrac23\)
Show working
- \(27^{\frac23}=9\)
- also \(\tfrac32n=9\Rightarrow n=6\)
Answer 9
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(a=27^{\frac23}\) | M1 | ||
| 9 | A1 | ||
| n = 6 | B1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? x
What do we know? reciprocal power \(-\tfrac23\)
Plan flip, cube root, square
Show working
- \(\left(\tfrac8{27}\right)^{\frac23}=\tfrac49\)
Answer \(\tfrac49\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| power \(-\tfrac23\) | M1 | ||
| \(\tfrac49\) | A1 | ||
| Total: 2 marks | |||
Can I do this?
- I can tell whether the unknown is in the power or in the base.
- I can solve \(x^n=k\) for whole, negative and fractional \(n\) using the reciprocal power.
- I know when to give \(\pm\) and when there is one answer.
- I can find \(a\) and \(n\) in identities like \((ax^n)^3=8x^{15}\), and write roots as powers.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
The multiple-choice and guided questions need JavaScript. The concept checks and exam practice below work without it.
Your progress on this note is saved when JavaScript is on.