MathsICSE Class 8 · Exponents and Powers

Solving Equations with an Unknown Exponent (ICSE Class 8 Maths)

Updated 17 October 2026 · Fact-checked

To solve an equation such as $3^{x-1} = 81$, write both sides as powers of the same prime base ($81 = 3^4$), then equate the exponents ($x - 1 = 4$, so $x = 5$). If sums of powers appear, take the lowest power as a common factor first.

Write both sides with the same base, then equate the exponents and solve the simple equation; finally check by substituting.

Illustration drawn for QuestionBuddy.

Understand Solving for an Unknown Exponent

An exponential equation has the unknown in the exponent. The key fact is: if $a \neq = 0, 1, - 1$ and $a^{m} = a^{n}$ , then $m = n$ . So the whole job is to bring both sides to the same base.

Step 1: choose a base. Factorise numbers into primes: $81 = 3^{4}$ , $125 = 5^{3}$ , $32 = 2^{5}$ , $27 = 3^{3}$ , $9 = 3^{2}$ , $\frac{1}{125} = 5^{- 3}$ .

Step 2: apply the laws to collapse each side to a single power, with the product law ( $a^{m} a^{n} = a^{m + n}$ ), quotient law and power-of-power law.

Step 3: equate exponents and solve the resulting linear equation.

Sums and differences of powers. When the equation has $2^{x + 3} + 2^{x}$ , you cannot add exponents. Instead take out the smallest power: $2^{x} (2^{3} + 1) = 2^{x} \times 9$ . This is the typical olympiad twist.

Same exponent, different bases. If $4^{n} \times 2 5^{n}$ appears, combine bases: $(4 \times 25)^{n} = 10 0^{n}$ .

Check. Substitute the answer back; a wrong exponent shows up at once. Growth problems (a culture that doubles or triples each hour) reduce to the same equation or to testing successive powers.

Solving for an Unknown Exponent: formulas to remember

Equal powers, equal bases: $a^{m} = a^{n} \Rightarrow m = n (a \neq = 0, 1, - 1)$; The base must match first.
Common factor of powers: $a^{x + k} + a^{x} = a^{x} (a^{k} + 1)$; Take out the lowest power.
Combining bases: $a^{n} b^{n} = (ab)^{n}$; Only when the exponents are equal.

How to solve Solving for an Unknown Exponent questions

Make the bases the same, then compare exponents.

1Write each number as a power of a prime.
2If there are sums of powers, take out the common (lowest) power.
3Use the laws to reach one power on each side with the same base.
4Equate the exponents and solve the linear equation.
5Substitute back to verify.

Solving for an Unknown Exponent in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Growth by tripling

Suppose a mould patch starts with 3 cells and the number triples every hour, so after $t$ hours there are $3 \times 3^{t} = 3^{t + 1}$ cells.

After how many whole hours does the count first exceed 1000?

Try it first, then show the solutionSolution

$3^{6} = 729$ and $3^{7} = 2187$ .
$t + 1 = 7$ , so $t = 6$ (at $t = 5$ there are only $729$ ).

Answer: 6 hours (2187 cells).

Memory hook: List successive powers until you pass the target.

Tricks to remember Solving for an Unknown Exponent

Short memory aids. Say each one aloud once, then use it on a question the same day.

Same base first
Never equate exponents until the bases are identical.
Use it when: All exponential equations.
Pull out the smallest power
In $2^{x + 3} + 2^{x}$ take out $2^{x}$ , leaving $2^{3} + 1$ .
Use it when: Sums of powers.

Solving for an Unknown Exponent: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Direct reading · 1 mark · MCQBoard level

If $2^{x} = 32$ , then $x$ is:
(a) 4 (b) 5 (c) 6 (d) 16

Try it first, then show the solutionSolution

$32 = 2^{5}$ , so $x = 5$ .

Answer: (b)

Example 2: Equal bases · 2 marks · Short answerBoard level

Solve $3^{x - 1} = 81$ .

Try it first, then show the solutionSolution

$81 = 3^{4}$ .
$x - 1 = 4$ , so $x = 5$ .
Check: $3^{4} = 81$ .

Answer: $x = 5$

Example 3: Reciprocal base · 3 marks · Short answerHarder board level

Solve $(\frac{2}{5})^{x} \times (\frac{2}{5})^{5} = (\frac{5}{2})^{- 9}$ .

Try it first, then show the solutionSolution

$(\frac{5}{2})^{- 9} = (\frac{2}{5})^{9}$ .
LHS $= (\frac{2}{5})^{x + 5}$ .
$x + 5 = 9$ , so $x = 4$ .

Answer: $x = 4$

Example 4: Sum of powers · 4 marks · Short answerHarder board level

Solve $2^{x + 3} + 2^{x} = 72$ .

Try it first, then show the solutionSolution

Take $2^{x}$ common: $2^{x} (2^{3} + 1) = 2^{x} \times 9$ .
$9 \times 2^{x} = 72 \Rightarrow 2^{x} = 8 = 2^{3}$ .
$x = 3$ . Check: $2^{6} + 2^{3} = 64 + 8 = 72$ .

Answer: $x = 3$

Example 5: Growth problem · 4 marks · Case-basedHOTS / competency

A culture begins with 5 cells and the count multiplies by 4 each hour, so after $t$ hours it is $5 \times 4^{t}$ . (i) Write the count after 3 hours. (ii) The count was 80 at some earlier time; find $t$ . (iii) Find the first whole hour when the count exceeds 5000.

Try it first, then show the solutionSolution

(i) $5 \times 4^{3} = 5 \times 64 = 320$ .
(ii) $5 \times 4^{t} = 80 \Rightarrow 4^{t} = 16 = 4^{2}$ , so $t = 2$ .
(iii) $4^{t} > 1000$ : $4^{4} = 256$ , $4^{5} = 1024$ . So $t = 5$ (count $5120$ ); at $t = 4$ the count is $1280$ .

Answer: (i) 320; (ii) $t = 2$ ; (iii) 5 hours.

Where students lose marks in Solving for an Unknown Exponent

The slips examiners see most often, and what to write instead.

Mistake: Equating exponents with different bases
$a^{m} = b^{n}$ does not mean $m = n$ unless the bases are equal.
Wrong working
$9^{x} = 3^{4} \Rightarrow x = 4$ .
Right working
$3^{2 x} = 3^{4} \Rightarrow x = 2$ .
Do this: Convert to a common prime base first.
Usually costs: Answer and method marks.
Mistake: Adding exponents in a sum
The product law applies to products, not to sums.
Wrong working
$2^{x + 3} + 2^{x} = 2^{2 x + 3}$ .
Right working
$2^{x + 3} + 2^{x} = 2^{x} \times 9$ .
Do this: Take the lowest power out as a common factor.
Usually costs: Most of the marks in the question.

Exam tips

Write the prime factorisation of each constant before starting; it takes seconds.
Always substitute your answer back into the original equation.
If the answer is a fraction such as $\frac{3}{2}$ , leave it as a fraction.

Practice questions on Solving for an Unknown Exponent

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If $5^{x} = \frac{1}{125}$ , then $x$ equals:
(a) 3 (b) $- 3$ (c) $\frac{1}{3}$ (d) $- \frac{1}{3}$
Hint
Write $\frac{1}{125}$ as $5^{- 3}$ .
Answer
(b)
Q2 · 2 marks · Short answerBoard level
Solve $7^{2 x} = 343$ .
Hint
$343 = 7^{3}$ .
Answer
$2 x = 3$ , so $x = \frac{3}{2}$ .
Q3 · 3 marks · Short answerHarder board level
Solve $9^{x} \times 3^{2} = 2 7^{x - 1}$ .
Hint
Make all three bases 3.
Answer
$3^{2 x} \times 3^{2} = 3^{3 (x - 1)}$ , so $2 x + 2 = 3 x - 3$ and $x = 5$ . Check: $9^{6} = 531441 = 2 7^{4}$ .
Q4 · 3 marks · Short answerHarder board level
Solve $5^{x} + 5^{x + 1} = 750$ .
Hint
Take 5^x common.
Answer
$5^{x} (1 + 5) = 750 \Rightarrow 5^{x} = 125 = 5^{3}$ , so $x = 3$ . Check: $125 + 625 = 750$ .
Q5 · 4 marks · Long answerHOTS / competency
Find $n$ if $4^{n} \times 2 5^{n} = 1 0^{6}$ , and evaluate $\frac{2 ^{n} \cdot 3 ^{n}}{6 ^{n - 2}}$ using that $n$ .
Hint
Combine the bases with the same exponent.
Answer
$4^{n} 2 5^{n} = 10 0^{n} = 1 0^{2 n} = 1 0^{6}$ , so $n = 3$ . Then $\frac{2 ^{3} \cdot 3 ^{3}}{6 ^{1}} = \frac{6 ^{3}}{6} = 6^{2} = 36$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Express 81 and 125 as powers of primes.Show answer
81 = 3^4 and 125 = 5^3.
2.If a^m = a^n, what follows?Show answer
m = n (for a not 0, 1, -1).
3.What is 2^(x+3) + 2^x in factorised form?Show answer
2^x x 9.
4.How do you combine 4^n x 25^n?Show answer
(4 x 25)^n = 100^n.

Solving for an Unknown Exponent: frequently asked questions

What if the bases cannot be made equal?

Class 8 questions are always designed so that they can; look for common primes or reciprocals.

Why can the base not be 1?

Because 1 to any power is 1, so equal results would not force equal exponents.