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

Power of a Power, Power of a Product, and Power of a Quotient

Power of a power

When a power is itself raised to another power, MULTIPLY the indices:

(a^m)^n = a^{mn}

For example, $(a^m)^n$ means $a^m$ multiplied by itself $n$ times, which is $a$ multiplied by itself $m\times n$ times in total.

Power of a product

When a PRODUCT of two bases is raised to a power, the power distributes to each factor:

(ab)^n = a^n b^n

Power of a quotient

Similarly, when a QUOTIENT is raised to a power, the power distributes to both numerator and denominator:

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \qquad b \neq 0

Note

Do not confuse $(a^m)^n$ with $a^{(m^n)}$

$(a^m)^n = a^{mn}$ always MULTIPLIES the two indices. It is a different (and much larger) quantity from $a$ raised to the power $m^n$ — the two expressions are only ever equal in special cases. Always read the brackets carefully to see exactly which quantity is being raised to which power.

Putting the five laws together …

Definition 1Power of a Power

$(a^m)^n = a^{mn}$ — when raising a power to another power, multiply …

Definition 2Power of a Product / Quotient

$(ab)^n = a^nb^n$ and $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ ( $b\neq0$ ) — the power distributes to every factor of a product, or to both numerator a …

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