Problem 19 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 25^{3 / 2} $$

Step-by-Step Solution

Answer

The value of \( 25^{3/2} \) is 125.

Step 1: Identify the Expression Format

The expression is given as \( 25^{3/2} \). This is an expression with an exponent that is a fraction. The fraction \( \frac{3}{2} \) means that we have both a square root and a power to evaluate.

Step 2: Break Down the Fractional Exponent

The fractional exponent \( \frac{3}{2} \) can be understood as taking the square root first (the denominator) and then raising it to the power of 3 (the numerator). This means we first calculate \( 25^{1/2} \) and then raise the result to the power of 3.

Step 3: Calculate the Square Root

Calculate \( 25^{1/2} \). The square root of 25 is 5, so \( 25^{1/2} = 5 \).

Step 4: Raise the Result to the Third Power

Now that we have \( 25^{1/2} = 5 \), we raise 5 to the power of 3 to find \( 5^3 \). Calculate \( 5^3 = 5 \times 5 \times 5 = 125 \).

Step 5: State the Final Result

The value of the expression \( 25^{3/2} \) is 125.

Key Concepts

Powers and Roots

When dealing with fractional exponents, understanding the relationship between powers and roots is crucial. Let's consider the expression \( 25^{3/2} \). The fraction \( \frac{3}{2} \) indicates two operations: a square root and an exponentiation.

The denominator of the fraction, which is 2, tells you to take the square root, and the numerator, which is 3, tells you to raise the result to the power of 3.

• The square root of 25 is 5 because \( 5 \times 5 = 25 \).
• After finding the square root, raise 5 to the power of 3, which results in \( 5 \times 5 \times 5 = 125 \).

This layered approach allows fractional exponents to transform complex exponentiation into simpler steps, utilizing both roots and powers.

Evaluating Expressions

Evaluating expressions like \( 25^{3/2} \) without a calculator enhances your understanding of mathematical relationships. The key is to decode the expression into simpler operations.

Here's how you can easily tackle it:

• Identify: Recognize that the expression has a fractional exponent.
• Break Down: Understand that it involves both a root and a power. For \( \frac{3}{2} \), split it into square rooting and then cubing.
• Compute: First, compute \( 25^{1/2} \) (square root of 25), giving you 5.
• Final Step: Raise 5 to the power of 3, resulting in 125.

This process sharpens your skills in evaluating complex algebraic expressions.

Exponent Rules

Knowing exponent rules is fundamental in simplifying and evaluating expressions. With fractional exponents, such as \( 25^{3/2} \), these rules become very handy.

Some key exponent rules include:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Fractional Exponent Understanding: \(a^{m/n} = \sqrt[n]{a^m}\)

Applying these rules, \( 25^{3/2} \) can be broken down into \((25^{1/2})^3\). Here, \( 25^{1/2} \) is the square root, and then we raise it to the 3rd power. Recognizing and applying these rules will greatly aid in simplifying and solving problems with exponents.