Problem 30 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{125}{8}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{25}{4} \).

Step 1: Understand the Exponent Fraction

The fraction in the exponent, \( \frac{2}{3} \), means we'll be taking the cube root of the base first, then squaring the result. So, \( \left(\frac{125}{8}\right)^{2/3} = \left(\left(\frac{125}{8}\right)^{1/3}\right)^2 \).

Step 2: Calculate the Cube Root

Find the cube root of both the numerator and the denominator separately: \( \sqrt[3]{125} = 5 \) because \( 5^3 = 125 \), and \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \). This gives us \( \sqrt[3]{\frac{125}{8}} = \frac{5}{2} \).

Step 3: Square the Result

Now, square the result from the previous step: \( \left(\frac{5}{2}\right)^2 = \frac{5^2}{2^2} = \frac{25}{4} \).

Step 4: Solution Verification

Verify each calculation step to ensure accuracy. Both cube roots and the subsequent squaring have been correctly performed as \( \frac{25}{4} \).

Key Concepts

Cube Roots

Have you ever wondered how to "unroll" a number that has been multiplied by itself three times? That's what cube roots help us do. The cube root of a number is a special value that, when multiplied by itself three times, gives us the original number. For example, the cube root of 125 is 5, because when you multiply 5 by itself twice more, you get back to 125:

• First step: 5 x 5 = 25
• Second step: 25 x 5 = 125

Similarly, the cube root of 8 is 2, because 2 times 2 times 2 equals 8:

• 2 x 2 = 4
• 4 x 2 = 8

Understanding this concept is important, especially when we work with fractional exponents like \( \frac{1}{3} \), which is another way of representing cube roots. Breaking down numbers in this way allows us to simplify and solve expressions more easily.

Exponentiation

Exponentiation is a powerful mathematical tool that essentially involves multiplying a number by itself a certain number of times, indicated by an exponent. This can be particularly perplexing with fractional exponents, such as \( \frac{2}{3} \). In these cases, the fraction provides a two-step instruction:

• The denominator (3) indicates taking the cube root.
• The numerator (2) indicates then squaring the result.

So, for our expression \( \left(\frac{125}{8}\right)^{2/3} \), the exponent \( \frac{2}{3} \) tells us to find the cube root of \( \frac{125}{8} \) first. After finding the cube root, we then square the outcome to arrive at the final result. This sequential process highlights the power of exponentiation, allowing us to transform complex numbers into something simpler and more manageable. It’s like following a recipe where you need to do each step in order for it to turn out just right.

Numerator and Denominator

In fractions, the numerator and the denominator are key players that define the whole value. The numerator, the number above the fraction line, indicates how many parts of a whole you have. The denominator, positioned below the line, tells you into how many total parts the whole is divided.When dealing with cube roots, as seen in the expression \( \frac{125}{8} \), each part can be treated individually. First, you take the cube root of the numerator: \( 125 \), equating to 5. Because 5 multiplies by itself three times to get 125. Then, handle the denominator: \( 8 \), resulting in 2, since 2 cubed returns to 8.

• Numerator's cube root: \( 5 \)
• Denominator's cube root: \( 2 \)

Post cube root extraction, dividing these gives \( \frac{5}{2} \), which is then squared to reach our final answer. Splitting operations between numerator and denominator simplifies otherwise complex calculations and helps maintain accuracy throughout solving.