An exponent outside parentheses tells you how many copies of everything inside to multiply. With that, the product rule does the rest.
Por qué funciona
$(x^{2})^{3} = x^{2} \cdot x^{2} \cdot x^{2}$: three groups of two factors, or $3 \cdot 2 = 6$ factors in all. So $(x^{2})^{3} = x^{6}$.
$(2x)^{3} = 2x \cdot 2x \cdot 2x$. Regroup the factors: $(2 \cdot 2 \cdot 2)(x \cdot x \cdot x) = 8x^{3}$. Every factor inside the parentheses is used three times, the 2 included.
$(2x)^{3} = 2x \cdot 2x \cdot 2x$. Regroup the factors: $(2 \cdot 2 \cdot 2)(x \cdot x \cdot x) = 8x^{3}$. Every factor inside the parentheses is used three times, the 2 included.
Idea clave
A power of a power multiplies the exponents: $(a^{m})^{n} = a^{mn}$. A power of a product raises every factor: $(ab)^{n} = a^{n}b^{n}$.
Worked examples
Ejemplo resuelto: Power of a power
Write $(5^{3})^{4}$ as one power.
- Four copies of $5^{3}$, each with 3 factors of 5.
- $4 \cdot 3 = 12$ factors in all.
Respuesta: $5^{12}$
Ejemplo resuelto: Power of a product
Simplify $(3y^{4})^{2}$.
- Square every factor inside: $3^{2} \cdot (y^{4})^{2}$.
- $3^{2} = 9$ and $(y^{4})^{2} = y^{8}$.
Respuesta: $9y^{8}$
Ejemplo resuelto: A negative inside
Simplify $(-2a^{2}b)^{3}$.
- Cube each factor: $(-2)^{3} \cdot (a^{2})^{3} \cdot b^{3}$.
- $(-2)^{3} = -8$, since three negative factors make a negative.
- $(a^{2})^{3} = a^{6}$, and $b$ becomes $b^{3}$.
Respuesta: $-8a^{6}b^{3}$
Error común
Raising only the letter: $(2x)^{3} = 2x^{3}$. The 2 is inside the parentheses, so it is cubed too. Check with $x = 1$: $(2 \cdot 1)^{3} = 8$, but $2 \cdot 1^{3} = 2$. Also, the rule is for products, not sums: $(x + y)^{2}$ is not $x^{2} + y^{2}$. Try $x = y = 1$: $(1 + 1)^{2} = 4$, but $1^{2} + 1^{2} = 2$.