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Powers of Powers

30 دقيقة

الشرح

🎯 What you will learn

A cube-shaped tank has edge $2^7$ cm, so its volume is $\left(2^7\right)^3$ — an exponent on top of an exponent. This lesson shows you how to collapse that into a single power instead of writing it all out.

Power of a power

Raising a power to another exponent, as in

(a^m)^n

: the base is

a

, the inner exponent is

m

and the outer exponent is

n

(a^{m})^{n}=a^{m\times n}

for any real

a\neq0

Where the law comes from

(2^3)^4=2^3\times2^3\times2^3\times2^3=2^{3+3+3+3}=2^{12}

Power of a product of powers

The outer exponent is shared out over each factor, multiplying the exponent of each one.

(a^{m}\times b^{n})^{k}=a^{mk}\times b^{nk}

Power of a quotient of powers

The outer exponent goes onto the numerator and the denominator alike, provided the denominator is not zero.

\left(\dfrac{a^{m}}{b^{n}}\right)^{k}=\dfrac{a^{mk}}{b^{nk}}

x⁵ · x⁻² ÷ xGiven (x ≠ 0)

Every line rests on a rule — read the rule beside it.

▶ A reminder: how exponents combine under multiplication and division.

Warning: no sharing over a sum

(a^{m}+b^{n})^{k}\neq a^{mk}+b^{nk}

Recall the earlier exponent laws

a^m\times a^n=a^{m+n}

a^m\div a^n=a^{m-n}

🧩 Worked examples

أمثلة محلولة

Worked exampleA power of a power

Simplify $(5^4)^2$ .

Worked exampleA negative exponent

Simplify $(7^{-2})^5$ .

Worked exampleA power of a product

Simplify $(2^3\times3)^4$ .

Worked exampleA power of a quotient

Simplify $\left(\dfrac{2^3}{7^2}\right)^5$ .