KJSEA 2025 Maths Q5 — Indices: Solving 2^X × 2^3 = 32

KJSEA 2025 Grade 9 Numbers

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The Question

“Kiroei picked a flash card with the equation 2 raised to the power X multiplied by 2 cubed equals 32. What is the value of X?”

1

Combine the powers on the left

The left side is a product of two powers that share the same base of 2. One law of indices says that when you multiply powers of the same base you keep the base and add the indices. So the two factors merge into a single power of 2 whose index is X plus 3.

2X×23=2X+32^{X} \times 2^{3} = 2^{X+3}
2

Write 32 as a power of 2

To compare both sides we need them written with the same base. Multiplying 2 by itself five times gives 32, so 32 is exactly 2 to the fifth power. This turns the right side into a power of 2 as well.

32=2×2×2×2×2=2532 = 2 \times 2 \times 2 \times 2 \times 2 = 2^{5}
3

Equate the indices

Now both sides are single powers of the same base 2. If two equal powers have the same base, their indices must be equal. So we can drop the base and simply set the index on the left equal to the index on the right.

2X+3=25⇒X+3=52^{X+3} = 2^{5} \Rightarrow X + 3 = 5
4

Solve for X

This leaves a simple linear equation. Subtracting 3 from both sides isolates X and gives the value picked on the flash card.

X=5−3=2X = 5 - 3 = 2

Final Result

X equals 2, which is answer C.

Why this method works

The method works because the laws of indices let us rewrite both sides of the equation as a single power of one common base. Powers of the same base are equal only when their exponents are equal, so once both sides read 2 to some power, comparing the equation collapses into comparing just the indices. That converts an intimidating exponential equation into an easy linear one that can be solved by ordinary arithmetic.

✓

Substitute back: 2^2 × 2^3 = 4 × 8 = 32, which matches the right side, so X = 2 is correct.