Simplify $\sqrt[4]{16} : 2^3$
Answer (2)
We simplified the expression 4 16 : 2 3 and found that it equals 4 1 . This involves calculating the fourth root of 16, evaluating 2 3 , and then simplifying the resulting fraction. The final answer is 4 1 .
;
Simplify the fourth root of 16: 4 16 = 2 .
Evaluate 2 3 : 2 3 = 8 .
Divide the results: 8 2 = 4 1 .
Express the result as a fraction: The simplified expression is 4 1 .
Explanation
Understanding the problem We are asked to simplify the expression 4 16 : 2 3 . This involves finding the fourth root of 16 and then dividing the result by 2 3 .
Simplifying the fourth root First, let's find the fourth root of 16, which can be written as 4 16 = 1 6 4 1 . Since 16 = 2 4 , we can rewrite this as ( 2 4 ) 4 1 . Using the power of a power rule, we get 2 4 ⋅ 4 1 = 2 1 = 2 .
Evaluating the power Next, we need to evaluate 2 3 , which is 2 ⋅ 2 ⋅ 2 = 8 .
Dividing the results Now, we divide the result of the fourth root by 2 3 , which is 2 3 4 16 = 8 2 .
Simplifying the fraction Finally, we simplify the fraction 8 2 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us 8 2 = 8 ÷ 2 2 ÷ 2 = 4 1 . We can also express this as a power of 2: 4 1 = 2 2 1 = 2 − 2 .
Final Answer Therefore, the simplified expression is 4 1 or 2 − 2 .
Examples
Understanding roots and exponents is crucial in many scientific fields. For example, in physics, the energy of a photon is related to its frequency by the equation E = h f , where h is Planck's constant. If you need to find the frequency f given the energy E , you would use the equation f = h E . Similarly, in finance, compound interest calculations involve exponents. If you invest a principal amount P at an annual interest rate r compounded n times per year, the amount A after t years is given by A = P ( 1 + n r ) n t . Simplifying expressions with roots and exponents is a fundamental skill that enables you to solve these types of problems efficiently.
