Problem 35 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 4^{-3 / 2} $$

Step-by-Step Solution

Answer

\( 4^{-3/2} = \frac{1}{8} \).

Step 1: Understand the Expression

The expression given is \( 4^{-3/2} \). This is a power expression with a fractional and negative exponent, which indicates the involvement of square roots and reciprocals.

Step 2: Rewrite the Negative Exponent

A negative exponent means we take the reciprocal of the base. Hence, \( 4^{-3/2} = \frac{1}{4^{3/2}} \). This step helps simplify the handling of exponents.

Step 3: Evaluate the Fractional Exponent

The expression now is \( \frac{1}{4^{3/2}} \). A fractional exponent like \( 3/2 \) can be rewritten as a power and a root: \( 4^{3/2} = (\sqrt{4})^3 \).

Step 4: Evaluate the Square Root

Find the square root of 4: \( \sqrt{4} = 2 \). This means \( (\sqrt{4})^3 \) becomes \( 2^3 \).

Step 5: Evaluate the Remaining Exponent

Calculate \( 2^3 = 8 \). Therefore, the expression \( 4^{3/2} \) evaluates to 8.

Step 6: Complete the Expression

Return to the reciprocal from Step 2: \( \frac{1}{4^{3/2}} = \frac{1}{8} \). This is the final evaluation of the expression.

Key Concepts

Negative Exponent

Negative exponents can seem tricky at first, but they follow a simple rule. When you see a negative exponent, like in the expression \( a^{-n} \), it means you should take the reciprocal of the base raised to the corresponding positive exponent. In simpler terms, \( a^{-n} = \frac{1}{a^n} \).

This transformation involves flipping the base to the denominator (or numerator if it’s already in that form). For example, if you encounter \( 4^{-3/2} \), rewrite it as \( \frac{1}{4^{3/2}} \). This allows you to handle the rest of the expression without worrying about the negative sign.

Remember these key points when working with negative exponents:

• A negative exponent indicates a reciprocal action.
• Always convert the negative exponent into a positive form.
• After flipping the base, continue to calculate as normal.

Fractional Exponent

Fractional exponents represent both roots and powers in a single expression. They can be split into two parts: the numerator and the denominator. The denominator introduces a root, while the numerator indicates the power. For instance, the expression \( a^{m/n} \) means the \( n \)-th root of \( a \) raised to the \( m \)-th power.

To simplify, take \( 4^{3/2} \) as an example. Here, the 2 in the denominator means you need the square root of 4, which is 2. The numerator, 3, tells you to raise that result to the third power: \( (\sqrt{4})^3 = 2^3 \). This process helps break down complex expressions step by step.

Some tips for working with fractional exponents:

• Identify the root from the denominator.
• Identify the power from the numerator.
• Apply the root first, then the power, or vice versa, depending on the context.

Square Roots

Square roots are fundamental in mathematics, providing a way to find a number that, when multiplied by itself, results in the original number. The square root of a number \( x \) is often denoted as \( \sqrt{x} \).

To see this in action, consider the number 4. Its square root is 2, because \( 2 \cdot 2 = 4 \). Understanding square roots is essential when dealing with fractional exponents. In \( 4^{3/2} \), for example, you first find the square root of 4, which simplifies the expression significantly.

Key aspects to remember about square roots:

• The square root represents a number that multiplied by itself gives the original number.
• They are commonly used in physics, geometry, and algebra to simplify expressions.
• When included in fractional exponents, they aid in breaking down complex expressions.