Answered

Select the correct answer.



Recall that $\sqrt[n]{x}=p$ means "the number $p$, such that $p^n=x$."



Using this definition, which solution correctly shows that $\sqrt[5]{6}=6^{\frac{1}{5}}$?


A$(6^{\frac{1}{5}})^5 = 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} = 6^{\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}} = 6^1 = 6$
B$(6^{\frac{1}{5}})^5 = 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} = 6\left(\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}\right) = 6 \cdot \frac{5}{5} = 6$
C$(6^{\frac{1}{5}})^5 = 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} = 5 \cdot 6^{\frac{1}{5}} = 5 \cdot \frac{1}{5} \cdot 6 = 6$
D$(6^{\frac{1}{5}})^5 = 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} \cdot 6^{\frac{1}{5}} = 6^{\frac{1}{5}+\frac{1}{5}+\frac{1}{5}} = 6^0 = 0$

Verified Answer :

  • Define the nth root: \(\sqrt[n]{x} = p \iff p^n = x\).
  • Apply the identity \((6^{1/5})^5 = 6^{(1/5) \cdot 5} = 6^1 = 6\).
  • Recognize that adding exponents during repeated multiplication leads to \(\sum_{i=1}^{5} \frac{1}{5} = \frac{5}{5} = 1\).
  • Identify the first option as the correct mathematical representation: \((6^{1/5})^5 = 6^{1/5} \cdot 6^{1/5} \cdot 6^{1/5} \cdot 6^{1/5} \cdot 6^{1/5} = 6^{1/5+1/5+1/5+1/5+1/5} = 6^{5/5} = 6^1 = 6\).
  • Final Answer: \(\boxed{\text{First Option}}\)

Explanation

  1. Understanding the Problem and Definitions

The problem asks us to verify that \(\sqrt[5]{6} = 6^{1/5}\) by using the definition of the \(n\)th root: \(\sqrt[n]{x} = p\) is equivalent to \(p^n = x\). Setting \(p = 6^{1/5}\) and \(n = 5\), we need to confirm whether \((6^{1/5})^5 = 6\). To evaluate this, we use properties of exponentiation, specifically that raising a power to a power involves multiplying the exponents (\((a^m)^n = a^{m \cdot n}\)) and that repeated multiplication of the same base involves adding exponents (\(a^m \cdot a^n = a^{m+n}\)).

  1. Evaluating the Correct Logical Path

We examine the first option:
\[(6^{1/5})^5 = 6^{1/5} \cdot 6^{1/5} \cdot 6^{1/5} \cdot 6^{1/5} \cdot 6^{1/5}\]
By the property of multiplying terms with the same base, we add the exponents:
\[6^{1/5+1/5+1/5+1/5+1/5} = 6^{5/5}\]
Since \(5/5 = 1\), the expression simplifies as:
\[6^{5/5} = 6^1 = 6\]
This result satisfies the definition \(\sqrt[n]{x} = p \implies p^n = x\), correctly demonstrating that \(6^{1/5}\) is indeed the fifth root of \(6\).

  1. Analyzing Alternative Options and Concluding

Comparing the other options:

  • The second option incorrectly simplifies \(6^{5/5}\) as \(6^{1/5}\).
  • The third option uses incorrect notation, implying the multiplication by 6 happens outside the exponent.
  • The fourth option incorrectly treats the exponentiation as multiplication by the power, which violates standard exponent rules.

Thus, the first option is the only one that correctly follows the algebraic steps to show that \((6^{1/5})^5 = 6\).

Examples

This concept is fundamental in scientific calculations and engineering. For example, when calculating the growth rate of an investment or population that compounds annually, you often need to find the equivalent monthly or daily rate. If an investment grows by a factor of 6 over 5 units of time, the growth per unit of time is found by taking the 5th root of 6, written as \(6^{1/5}\). Understanding how to manipulate these fractional exponents allows for easier calculation and comparison of different growth scenarios in finance and biology.

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