What is the simplified value of the exponential expression $27^{\frac{1}{3}}$?
A. $\frac{1}{3}$
B. $\frac{1}{9}$
C. 3
D. 9
Answer (1)
Recognize that 2 7 3 1 is the same as the cube root of 27.
Find the number that, when multiplied by itself three times, equals 27.
Determine that 3 × 3 × 3 = 27 .
Conclude that the simplified value of the expression is 3 .
Explanation
Understanding the problem We are asked to find the simplified value of the exponential expression 2 7 3 1 . This expression means '27 raised to the power of 3 1 '.
Rewriting the expression Recall that raising a number to the power of 3 1 is the same as taking the cube root of that number. In mathematical terms, 2 7 3 1 = 3 27 .
Finding the cube root Now, we need to find a number that, when multiplied by itself three times, equals 27. Let's try a few numbers:
1 × 1 × 1 = 1 (too small)
2 × 2 × 2 = 8 (too small)
3 × 3 × 3 = 27 (correct!)
Final Answer Since 3 × 3 × 3 = 27 , the cube root of 27 is 3. Therefore, 2 7 3 1 = 3 .
Examples
Imagine you are designing a cube-shaped box that needs to hold exactly 27 cubic inches of volume. To find the length of each side of the box, you would need to calculate the cube root of 27, which is 2 7 3 1 . This tells you that each side of the cube should be 3 inches long. Understanding exponential expressions and roots is useful in various fields like engineering, architecture, and even everyday tasks like packing and storage.
