Business Mathematics and Basic Statistics · Ch 3 — Laws of Indices

Simplifying Expressions and Solving Simple Exponential Equations

Simplifying an expression

To simplify an expression involving indices, apply the laws one at a time, in a clear, deliberate order: resolve any powers-of-powers first, then combine products/quotients of the same base using the product/quotient laws, and finally express the result with POSITIVE indices only (using the negative-exponent law) unless the question explicitly asks otherwise.

Solving a simple exponential equation

An exponential equation has the unknown in the exponent (e.g. $2^x = 32$ ). The standard technique used throughout this chapter's scope is to express BOTH sides as powers of the SAME base, then equate the exponents directly:

\text{if } a^x = a^y \ (a>0,\ a\neq1), \text{ then } x = y

For example, to solve $2^x=32$ : recognise $32=2^5$ , so $2^x=2^5$ , giving $x=5$ directly — no further calculation needed once both sides share a base.

Note

Finding the common base is the whole trick

Every simple exponential equation in this chapter's scope is solvable once both sides are rewritten with the SAME base (a prime number is usually the easiest common base to look for, e.g. rewriting 81 as $3^4$ or 32 as $2^5$ ). If a genuine common base cannot be found by inspection, the equation is beyond the algebraic ("equate the exponents") scope of this chapter.

A worked simplification pattern …

Definition 1Exponential Equation

An equation in which the unknown appears in the exponent, e.g. $2^x=32$ . Solved (within this chapter's scope) by rewriting both sides as powers of the SAME base and equating the exp …

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