Problem 130 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using my calculator, I determined that \(6^{7}=279,936,\) so 6 must be a seventh root of \(279,936\).

Step-by-Step Solution

Answer

The statement does make sense as \(6^7\) is indeed \(279,936\), hence 6 is the seventh root of \(279,936\).

Step 1: Understanding the statement

The statement says that \(6^7 = 279,936\), and so, 6 must be the seventh root of 279,936, which means \(6 = \sqrt[7]{279,936}\). The given statement seems to make sense as powers and roots are inverse of each other

Step 2: Verifying the statement

Calculate the seventh root of 279,936. If the result is equal to 6, then the initial claim is correct. This verifies mathematical relation that exponentiation and finding roots are inverse operations of each other. Using a calculator, the seventh root of 279,936 is indeed equal to 6.

Step 3: Conclusion

After performing the operation, it is concluded that the statement is correct. 6 is indeed the seventh root of 279,936 as per the definition and properties of roots and exponentiation in mathematics, verifying the statement by calculations