KCSE 2025 Maths P1 Q3 — Solving an Indices Equation

KCSE 2025 Form 4 Algebra

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The Question

“Solve for x in the equation 4 raised to the power 3x, all multiplied by 8, equals one over 32, all raised to the power (2x minus 3).”

1

Spot the common base

Notice that every number in the equation — 4, 8 and 32 — is a power of 2. If we rewrite the whole equation using the single base 2, we will later be able to compare the powers directly instead of juggling different numbers.

4=22,8=23,32=254 = 2^{2}, \quad 8 = 2^{3}, \quad 32 = 2^{5}
2

Rewrite the left-hand side

Replace 4 with 2 squared and 8 with 2 cubed. Because a power raised to another power means you multiply the indices, the 3x multiplies with the 2. Then, since multiplying powers of the same base means you add the indices, the two exponents combine into one.

43x×8=(22)3x×234^{3x} \times 8 = (2^{2})^{3x} \times 2^{3}
=26x×23=26x+3= 2^{6x} \times 2^{3} = 2^{6x + 3}
3

Rewrite the right-hand side

One over 32 is one over 2 to the power 5, and a factor on the bottom moves to the top when we make its power negative, so it becomes 2 to the power minus 5. Raising that to the bracket (2x minus 3) means multiplying the indices, so expand minus 5 across the bracket.

132=2−5\frac{1}{32} = 2^{-5}
(2−5)2x−3=2−5(2x−3)=2−10x+15\left(2^{-5}\right)^{2x-3} = 2^{-5(2x-3)} = 2^{-10x + 15}
4

Compare the indices

Both sides are now single powers of 2 that are equal to each other. When two equal powers share the same base, their indices must themselves be equal, so we can drop the base and just set the exponents equal.

26x+3=2−10x+152^{6x + 3} = 2^{-10x + 15}
6x+3=−10x+156x + 3 = -10x + 15
5

Solve the linear equation

Gather the x terms on one side and the constants on the other by adding 10x to both sides and subtracting 3 from both sides. Combine like terms, then divide by the coefficient of x and simplify the fraction to lowest terms.

6x+10x=15−36x + 10x = 15 - 3
16x=12⇒x=1216=3416x = 12 \Rightarrow x = \frac{12}{16} = \frac{3}{4}

Final Result

x = 3/4.

Why this method works

The method works because exponentiation with a fixed base is a one-to-one relationship: if two powers of 2 are equal, the only way that can happen is for their exponents to be equal. Rewriting 4, 8 and 32 as powers of 2 forces both sides onto that single base, so the messy equation collapses into a simple linear equation in x. The index laws we lean on — multiply indices for a power of a power, add indices when multiplying, and negate the index to move a term across the fraction line — are just bookkeeping that keeps each expression equal to the original while pushing everything toward base 2.

✓

Substituting x = 3/4 gives left index 6(3/4) + 3 = 7.5 and right index -10(3/4) + 15 = 7.5, so both sides equal 2 to the power 7.5, confirming the solution.