Question

algebra unit 2 quick quiz #2
lessons 4 - 7
rn.1, rn.2
name:
date:

1. what is the correct exponential form of the following expression?

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b.
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1. use and the product rule and the definition of roots to explain the meaning of.

explanation:

Explanation:

Step1: Apply the product rule of exponents

The product rule states that \(a^m\times a^n=a^{m + n}\). For \(2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\), we have \(m=n=\frac{1}{5}\) and there are 5 terms. So, \(2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}=2^{\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}}\).

Step2: Simplify the exponent

\(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=\frac{1 + 1+1+1+1}{5}=\frac{5}{5}=1\). So, \(2^{\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}}=2^1 = 2\).

Step3: Relate to the definition of roots

By the definition of \(a^{\frac{1}{n}}=\sqrt[n]{a}\), if we let \(a = 2\) and \(n = 5\), then \(2^{\frac{1}{5}}\) is the fifth - root of 2. When we multiply \(2^{\frac{1}{5}}\) by itself 5 times (using the product rule \(a^m\times a^n=a^{m + n}\)), we get back to 2.

Answer:

\(2^{\frac{1}{5}}\) is the fifth - root of 2. When we use the product rule \(a^m\times a^n=a^{m + n}\) for \(2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\cdot2^{\frac{1}{5}}\), the exponents add up (\(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=1\)) and \(2^1 = 2\). So, \(2^{\frac{1}{5}}\) is a number which when multiplied by itself 5 times gives 2, which is the definition of the fifth - root of 2.