Power of a Power

When a power is raised to another power, you **keep the base and multiply the exponents**: $$(a^m)^n = a^{m \cdot n}$$ **Why it works:** the outer exponent $n$ says "use $a^m$ as a factor $n$ times". Then the product rule (add the exponents) does the rest: $$(2^3)^2 = 2^3 \cdot 2^3 = 2^{3+3} = 2^6 = 64$$ Adding the exponent $m$ a total of $n$ times is the same as $m \cdot n$. **Two related rules** come from the same idea: - **Power of a product:** $(ab)^n = a^n b^n$. Every factor inside the brackets gets the exponent, also the number in front: $(3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6$ - **Power of a quotient:** $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ (with $b \neq 0$) **Watch the brackets:** $(2^3)^2 = 2^6 = 64$, but $2^{3^2} = 2^9 = 512$. Without brackets, the exponent at the top is worked out first. In this lesson all exponents are positive whole numbers.

A Power of a Power

Simplify $(5^2)^4$.

Find the base and the two exponents: Base $5$, inner exponent $2$, outer exponent $4$ = A power raised to a power

Write it as a product: $(5^2)^4 = 5^2 \cdot 5^2 \cdot 5^2 \cdot 5^2 = 5^{2+2+2+2}$ = $5^8$

Use the shortcut: multiply the exponents: $2 \cdot 4 = 8$ = $(5^2)^4 = 5^8$

Answer: $(5^2)^4 = 5^8 = 390\,625$

Power of a Product

Simplify $(2x)^4$.

Give the exponent to every factor in the brackets: $(2x)^4 = 2^4 \cdot x^4$ = Both $2$ and $x$ get the exponent $4$

Work out the number: $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$ = $16$

Write the result: $16 \cdot x^4$ = $16x^4$

Answer: $(2x)^4 = 16x^4$

Don't Forget the Coefficient

Simplify $(3x^2)^3$.

Give the exponent to every factor: $(3x^2)^3 = 3^3 \cdot (x^2)^3$ = Two factors: $3$ and $x^2$

Raise the coefficient: $3^3 = 27$ = $27$

Power of a power for the variable: $(x^2)^3 = x^{2 \cdot 3} = x^6$ = $x^6$

Put the parts together: $27 \cdot x^6$ = $27x^6$

Answer: $(3x^2)^3 = 27x^6$

Power of a Quotient

Simplify $\left(\frac{x^3}{2}\right)^4$.

Raise the numerator and the denominator: $\left(\frac{x^3}{2}\right)^4 = \frac{(x^3)^4}{2^4}$ = Two separate powers

Numerator: multiply the exponents: $(x^3)^4 = x^{3 \cdot 4} = x^{12}$ = $x^{12}$

Denominator: work out the power: $2^4 = 16$ = $16$

Write the fraction: $\frac{x^{12}}{16}$ = $\frac{x^{12}}{16}$

Answer: $\left(\frac{x^3}{2}\right)^4 = \frac{x^{12}}{16}$

All Three Rules Together

Simplify $\frac{(2a^3)^2 \cdot a^4}{(a^2)^3}$.

Power of a product in the numerator: $(2a^3)^2 = 2^2 \cdot a^{3 \cdot 2} = 4a^6$ = $4a^6$

Product rule: add the exponents: $4a^6 \cdot a^4 = 4a^{6+4}$ = $4a^{10}$

Power of a power in the denominator: $(a^2)^3 = a^{2 \cdot 3}$ = $a^6$

Quotient rule: subtract the exponents: $\frac{4a^{10}}{a^6} = 4a^{10-6}$ = $4a^4$

Answer: $\frac{(2a^3)^2 \cdot a^4}{(a^2)^3} = 4a^4$

The power of a power rule turns long chains of multiplication into one quick step: - **Areas and volumes:** a square with side $10^3$ m has area $(10^3)^2 = 10^6$ m², and a cube with edge $2^2$ cm has volume $(2^2)^3 = 2^6 = 64$ cm³ - **Computer storage:** a kilobyte is $2^{10}$ bytes, and $(2^{10})^3 = 2^{30}$ bytes is a little more than one billion bytes - **Science:** in $E = mc^2$ the speed of light $3 \cdot 10^8$ m/s is squared: $(3 \cdot 10^8)^2 = 9 \cdot 10^{16}$ - **Algebra:** a square with side $3x^2$ has area $(3x^2)^2 = 9x^4$ Together with the product rule and the quotient rule, it lets you simplify almost any expression with powers of the same base.

Mistake: Adding the exponents: $(x^3)^4 = x^7$

Why: Adding is the product rule, for $x^3 \cdot x^4$. Here the outer $4$ means four copies of $x^3$, so the exponent is $3 + 3 + 3 + 3 = 12$.

Correct: Multiply the exponents: $(x^3)^4 = x^{3 \cdot 4} = x^{12}$

Mistake: Thinking $(a + b)^2 = a^2 + b^2$

Why: The rule $(ab)^n = a^n b^n$ is for a product, not a sum. Test with numbers: $(3 + 4)^2 = 7^2 = 49$, but $3^2 + 4^2 = 9 + 16 = 25$.

Correct: $(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2$, so $(3 + 4)^2 = 9 + 24 + 16 = 49$

Mistake: Treating $(2^3)^2$ and $2^{3^2}$ as the same

Why: The brackets change the order. $(2^3)^2$ squares $2^3$, while $2^{3^2}$ first works out the top exponent $3^2 = 9$.

Correct: $(2^3)^2 = 2^6 = 64$, but $2^{3^2} = 2^9 = 512$

Mistake: Forgetting the coefficient: $(3x^2)^3 = 3x^6$

Why: The exponent applies to every factor inside the brackets, and the number $3$ is a factor too.

Correct: $(3x^2)^3 = 3^3 \cdot x^6 = 27x^6$

Computer Storage

Computers count memory in powers of 2. Each bigger unit is $2^{10} = 1024$ times the one before it.

A kilobyte is $2^{10}$ bytes, a megabyte is $2^{10}$ kilobytes, so a megabyte is $(2^{10})^2 = 2^{20}$ bytes.

Enlarging a Photo

When a picture is enlarged, its width and height are both multiplied by the same factor, so its area (the number of pixels) grows by that factor squared.

Doubling the width and the height of a photo multiplies the number of pixels by $2^2 = 4$.

Energy and the Speed of Light

Einstein's formula $E = mc^2$ squares the speed of light $c$, which is about $3 \cdot 10^8$ meters per second.

To square $3 \cdot 10^8$, square the $3$ and the power of ten separately, because both are factors.

**Power of a power:** $(a^m)^n = a^{m \cdot n}$, keep the base and multiply the exponents

The rule comes from the product rule: $(2^3)^2 = 2^3 \cdot 2^3 = 2^{3+3} = 2^6$

**Power of a product:** $(ab)^n = a^n b^n$, every factor gets the exponent, including the coefficient: $(3x^2)^3 = 27x^6$

**Power of a quotient:** $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, raise the numerator and the denominator

Brackets matter: $(2^3)^2 = 64$, but $2^{3^2} = 512$

The rules work for products and quotients only: $(a + b)^2 \neq a^2 + b^2$

Q: Why do we multiply the exponents instead of adding them?

A: Because $(a^m)^n$ means $n$ copies of $a^m$ multiplied together. By the product rule their exponents add up: $m + m + \dots + m$ ($n$ times), which is $m \cdot n$.

Q: Is $(a^2)^3$ the same as $(a^3)^2$?

A: Yes. Both equal $a^6$, because $2 \cdot 3 = 3 \cdot 2$. Check with numbers: $(2^2)^3 = 4^3 = 64$ and $(2^3)^2 = 8^2 = 64$.

Q: What does $2^{3^2}$ mean without brackets?

A: A tower of exponents is worked out from the top down: first $3^2 = 9$, then $2^9 = 512$. If you mean $(2^3)^2 = 64$, you must write the brackets.

Q: Does the rule also work for a sum, like $(x + 2)^2$?

A: No. The exponent can be shared out only over factors. For a sum, multiply it out: $(x + 2)^2 = (x + 2)(x + 2) = x^2 + 4x + 4$.

Exit Presentation

Power of a Power

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Arithmetic / Exponents & Roots

Power of a Power

Learn why a power raised to a power means multiplying the exponents, and how to raise products, quotients and coefficients to a power.

Intermediate25 min

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