What is the following product?
[tex]$\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$[/tex]
Answer (2)
The product of four fourth roots of 7 is calculated by rewriting each fourth root as an exponent and then summing the exponents. This leads to the final result of 7, as 7 1 = 7 . So, the answer is 7.
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Rewrite the fourth root of 7 using exponents: 4 7 = 7 4 1 .
Express the product using exponents: 7 4 1 × 7 4 1 × 7 4 1 × 7 4 1 .
Add the exponents: 4 1 + 4 1 + 4 1 + 4 1 = 1 .
Simplify the expression: 7 1 = 7 .
Explanation
Understanding the Problem We are asked to find the product of four identical fourth roots of 7. This means we need to multiply 4 7 by itself four times.
Rewriting with Exponents We can rewrite the fourth root of 7 using exponents. Recall that n a = a n 1 . Therefore, 4 7 = 7 4 1 .
Substituting Exponents Now we can rewrite the original expression as: 7 4 1 × 7 4 1 × 7 4 1 × 7 4 1 .
Adding Exponents When multiplying numbers with the same base, we add the exponents. In this case, the base is 7, and we are adding 4 1 + 4 1 + 4 1 + 4 1 .
Calculating the Sum The sum of the exponents is 4 1 + 4 1 + 4 1 + 4 1 = 4 4 = 1 .
Final Result Therefore, the expression simplifies to 7 1 , which is simply 7.
Examples
Understanding roots and exponents is crucial in many scientific fields. For example, in physics, when dealing with wave functions or quantum mechanics, you often encounter expressions involving roots and exponents. Simplifying these expressions correctly, as we did here, allows for accurate calculations and predictions about the behavior of physical systems. Similarly, in finance, compound interest calculations rely heavily on understanding exponents, where the principal amount grows exponentially over time.
