Alliance Form 3 P1 Q6 — Laws of Indices (Simplify with Fractional Powers)

KCSE (Form 3) Form 3 Algebra

Published

The Question

“Simplify the expression 243 raised to the power minus two-fifths, multiplied by 125 raised to the power two-thirds, divided by 9 raised to the power minus three-halves, leaving your answer as a whole number.”

1

Write every base as a prime power

The numbers look awkward until you notice each one is a small prime raised to a power. Rewriting 243, 125 and 9 in prime form lets the laws of indices do the heavy lifting, because then the whole expression sits on just two bases, 3 and 5.

243=35125=539=32243 = 3^{5} \qquad 125 = 5^{3} \qquad 9 = 3^{2}
2

Apply the power-of-a-power rule

When a power is raised to another power you multiply the two indices. Doing this to each factor clears away the fractional exponents and turns everything into a simple whole-number index.

(35)−25=35×−25=3−2\left(3^{5}\right)^{-\frac{2}{5}} = 3^{5 \times -\frac{2}{5}} = 3^{-2}
(53)23=53×23=52\left(5^{3}\right)^{\frac{2}{3}} = 5^{3 \times \frac{2}{3}} = 5^{2}
(32)−32=32×−32=3−3\left(3^{2}\right)^{-\frac{3}{2}} = 3^{2 \times -\frac{3}{2}} = 3^{-3}
3

Turn the division into a subtraction of indices

Dividing by a power with the same base means subtracting its index. Subtracting a negative index minus three is the same as adding three, so dividing by 3 to the minus 3 becomes multiplying by 3 to the plus 3.

3−2×52÷3−3=3−2×52×333^{-2} \times 5^{2} \div 3^{-3} = 3^{-2} \times 5^{2} \times 3^{3}
4

Combine powers of the same base

The two factors that share the base 3 can be joined by adding their indices. Once combined, the base 3 has index 1, which is just 3 itself.

3−2×33=3−2+3=313^{-2} \times 3^{3} = 3^{-2+3} = 3^{1}
31×52=3×253^{1} \times 5^{2} = 3 \times 25
5

Evaluate the final product

With the indices sorted out, only a simple multiplication remains to give the whole-number answer the question asked for.

3×25=753 \times 25 = 75

Final Result

The expression simplifies to 75.

Why this method works

The method works because every number here is secretly a single prime raised to a power, so the entire problem lives on just two bases. The power-of-a-power rule (multiply the indices) removes the scary fractional exponents cleanly, since each fraction is designed to cancel with the base's exponent. Once each factor is a whole-number power, division simply becomes subtraction of indices and multiplication becomes addition of indices — but only for powers that share a base. Grouping the base-3 terms collapses them to 3 to the power 1, and what is left is an ordinary arithmetic product.

✓

Check: 3^{-2} = 1/9, 5^{2} = 25, and multiplying by 3^{3} = 27 gives (25 imes 27)/9 = 675/9 = 75.