Find the value of $\sqrt[3]{8^3 \times \sqrt{\frac{1}{64}}}$
Answer (1)
Simplify the square root: 64 1 = 8 1 .
Substitute back into the original expression: 3 8 3 × 8 1 .
Simplify the expression inside the cube root: 8 3 × 8 1 = 64 .
Evaluate the cube root: 3 64 = 4 . The final answer is 4 .
Explanation
Understanding the Problem We are asked to find the value of the expression 3 8 3 × \tSqrt 64 1 . This involves cube root, square root, exponentiation and a fraction.
Simplifying the Square Root First, let's simplify the square root term: 64 1 . Since 64 = 8 2 , we have 64 1 = 8 2 1 = 8 1 .
Substituting Back Now, substitute this value back into the original expression: 3 8 3 × 8 1 .
Simplifying the Expression We can simplify 8 3 × 8 1 as 8 3 × 8 − 1 = 8 3 − 1 = 8 2 = 64 . So the expression becomes 3 64 .
Finding the Cube Root Finally, we need to find the cube root of 64. Since 4 3 = 4 × 4 × 4 = 16 × 4 = 64 , we have 3 64 = 4 .
Final Answer Therefore, the value of the expression 3 8 3 × \tSqrt 64 1 is 4.
Examples
Understanding roots and exponents is crucial in many scientific fields. For example, calculating the volume of a cube involves cubing the side length, and finding the side length from a given volume involves taking the cube root. Similarly, calculating areas of squares and circles involves squaring and square roots. These concepts are also used in financial calculations, such as compound interest, where exponents determine the growth of investments over time.
