第1章前提知識Prerequisites

Finding the Power of a Product積の累乗を求める

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider ${(p q)}^{3} .$ We begin by using the associative and commutative properties of multiplication to regroup the factors.

二つの指数の式の積の累乗を簡単にするには、指数の積の累乗の規則を使う。これは、因数の積の累乗を、各因数の累乗の積に分ける規則である。たとえば ${(p q)}^{3}$ を考えよう。まず乗法の結合法則と交換法則を使って因数を組み替える。

\begin{matrix} {(p q)}^{3} & = & \overset{3 factors}{\overset{}{(p q) \cdot (p q) \cdot (p q)}} \\ = & p \cdot q \cdot p \cdot q \cdot p \cdot q \\ = & \overset{3 factors}{\overset{}{p \cdot p \cdot p}} \cdot \overset{3 factors}{\overset{}{q \cdot q \cdot q}} \\ = & p^{3} \cdot q^{3} \end{matrix}

In other words, ${(p q)}^{3} = p^{3} \cdot q^{3} .$

言い換えると ${(p q)}^{3} = p^{3} \cdot q^{3}$ である。

The Power of a Product Rule of Exponents 積の累乗の法則

For any real numbers $a$ and $b$ and any integer $n,$ the power of a product rule of exponents states that

どの実数 $a$ 、 $b$ と、どの整数 $n$ についても、指数の積の累乗の規則は次を述べる。

{(a b)}^{n} = a^{n} b^{n}

Example 7例7

Using the Power of a Product Rule積の累乗の規則を使う

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

積の累乗の規則を使って、次の各積をできるだけ簡単にせよ。答えは正の指数で書け。

ⓐ ${(a b^{2})}^{3}$ ⓑ ${(2 t)}^{15}$ ⓒ ${(−2 w^{3})}^{3}$ ⓓ $\frac{1}{{(−7 z)}^{4}}$ ⓔ ${(e^{−2} f^{2})}^{7}$

Solution 解答

Use the product and quotient rules and the new definitions to simplify each expression.

積と商の規則、および新しい定義を使って各式を簡単にする。

ⓐ ${(a b^{2})}^{3} = {(a)}^{3} \cdot {(b^{2})}^{3} = a^{1 \cdot 3} \cdot b^{2 \cdot 3} = a^{3} b^{6}$ ⓑ ${(2 t)}^{15} = {(2)}^{15} \cdot {(t)}^{15} = 2^{15} t^{15} = 32, 768 t^{15}$ ⓒ ${(−2 w^{3})}^{3} = {(−2)}^{3} \cdot {(w^{3})}^{3} = −8 \cdot w^{3 \cdot 3} = −8 w^{9}$ ⓓ $\frac{1}{{(−7 z)}^{4}} = \frac{1}{{(−7)}^{4} \cdot {(z)}^{4}} = \frac{1}{2, 401 z^{4}}$ ⓔ ${(e^{−2} f^{2})}^{7} = {(e^{−2})}^{7} \cdot {(f^{2})}^{7} = e^{−2 \cdot 7} \cdot f^{2 \cdot 7} = e^{−14} f^{14} = \frac{f^{14}}{e^{14}}$

Try It #7やってみよう7

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

積の累乗の規則を使って、次の各積をできるだけ簡単にせよ。答えは正の指数で書け。