Power of a product rule

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When a whole product is raised to a power, that power lands on each factor.
So

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

.
Every part inside the brackets is raised to the power.

Do not forget the number in front.
In

(2x)^3

, both

2

and

x

are cubed:

2^3 \times x^3 = 8x^3

.
A common mistake is to cube only the

x

and leave the

2

alone.

The rule extends to factors that already have exponents.
For

(2x^2)^2

, raise each part:

2^2 \times (x^2)^2 = 4x^4

, using

(x^a)^b = x^{ab}

.
So

(x^2)^2 = x^{2 \times 2} = x^4

It also works with fractional powers, which are roots.
For

(2x^6)^{\frac{1}{2}}

, take each factor to the

\frac{1}{2}

2^{\frac{1}{2}} \times (x^6)^{\frac{1}{2}}

.
Since

(x^6)^{\frac{1}{2}} = x^{6 \times \frac{1}{2}} = x^3

, this is

2^{\frac{1}{2}} x^3

Worked examples

Simplify

(3x)^2

.
Raise each factor to the power:

3^2 \times x^2

.
So

(3x)^2 = 9x^2

Simplify

(2x^2)^2

.
Raise each factor:

2^2 \times (x^2)^2

.
Since

(x^2)^2 = x^4

, this is

4x^4

Simplify

(2x^6)^{\frac{1}{2}}

.
Raise each factor to

\frac{1}{2}

2^{\frac{1}{2}} \times (x^6)^{\frac{1}{2}}

.
Since

(x^6)^{\frac{1}{2}} = x^3

, this is

2^{\frac{1}{2}} x^3

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