Simplify $\sqrt[4]{16} : 2^3$
Answer (2)
To simplify 4 16 : 2 3 , we find that 4 16 = 2 and 2 3 = 8 , resulting in 8 2 = 4 1 . The final answer is 4 1 .
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Calculate the fourth root of 16: 4 16 = 2 .
Calculate 2 cubed: 2 3 = 8 .
Substitute the values into the expression: 8 2 .
Simplify the fraction: 8 2 = 4 1 .
The simplified expression is 4 1 .
Explanation
Understanding the problem We are asked to simplify the expression 4 16 : 2 3 . This means we need to divide the fourth root of 16 by 2 raised to the power of 3. Let's break it down step by step.
Calculating the fourth root of 16 First, let's find the fourth root of 16. The fourth root of a number x is a value that, when raised to the fourth power, equals x . In this case, we are looking for a number that, when raised to the fourth power, equals 16. We know that 2 4 = 2 × 2 × 2 × 2 = 16 . Therefore, 4 16 = 2 .
Calculating 2 cubed Next, let's calculate 2 3 . This means 2 raised to the power of 3, which is 2 × 2 × 2 = 8 . So, 2 3 = 8 .
Substituting the values Now, we can rewrite the original expression 4 16 : 2 3 as 2 3 4 16 . Substituting the values we found, we get 8 2 .
Simplifying the fraction Finally, let's simplify the fraction 8 2 . Both the numerator and the denominator are divisible by 2. Dividing both by 2, we get 8 ÷ 2 2 ÷ 2 = 4 1 . Therefore, the simplified expression is 4 1 .
Examples
Understanding how to simplify expressions with roots and exponents is useful in many areas, such as calculating growth rates or dealing with scaling factors in geometry. For example, if you are designing a square garden and want to increase its area by a factor of 16, you need to understand roots to determine how much to increase the side length. Similarly, exponents are used in compound interest calculations to determine how investments grow over time. These concepts are fundamental in science, engineering, and finance.
