a) Find the value of $\sqrt[3]{8^3 \times \sqrt{\frac{1}{64}}}$
Answer (1)
Evaluate the square root: 64 1 = 8 1 .
Substitute into the original expression: 3 8 3 × 8 1 .
Simplify inside the cube root: 8 3 × 8 1 = 64 .
Evaluate the cube root: 3 64 = 4 , so the final answer is 4 .
Explanation
Understanding the Problem We are asked to find the value of the expression 3 8 3 × 64 1 . This involves cube roots, square roots, multiplication, and fractions.
Evaluating the Square Root First, let's evaluate the square root: 64 1 = 64 1 = 8 1 .
Substituting Back Now, substitute this result back into the original expression: 3 8 3 × 8 1 .
Simplifying the Expression Next, simplify the expression inside the cube root: 8 3 × 8 1 = 8 8 3 = 8 2 = 64 .
Evaluating the Cube Root Finally, evaluate the cube root: 3 64 = 4 .
Final Answer Therefore, the value of the expression 3 8 3 × 64 1 is 4.
Examples
Understanding roots and exponents is crucial in many fields, such as physics and engineering. For example, when calculating the volume of a cube with side length s , the volume V is given by V = s 3 . Conversely, if you know the volume and need to find the side length, you would take the cube root: s = 3 V . Similarly, square roots are used in calculating distances using the Pythagorean theorem. These concepts are also fundamental in financial calculations, such as determining the rate of return on an investment.
