Problem 61 - Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{24}{(2 \sqrt{x})^{3}} $$

Step-by-Step Solution

Answer

The expression in power form is \(3x^{-3/2}\).

Step 1: Simplify the Denominator

First, let's simplify the expression inside the parentheses in the denominator. The expression is \( (2 \sqrt{x})^{3} \), which can be expanded using \( (ab)^n = a^n b^n \). Applying this, we get: \( (2 \sqrt{x})^{3} = 2^{3} \cdot (\sqrt{x})^{3} = 8x^{3/2} \). This uses the fact that \( (\sqrt{x})^3 = (x^{1/2})^3 = x^{3/2} \).

Step 2: Write the Fraction in Power Form

Now substitute the simplified denominator back into the original fraction: \( \frac{24}{8x^{3/2}} \). This can be rewritten as \( \frac{24}{8} \cdot x^{-3/2} \). Simplifying \( \frac{24}{8} \) gives \( 3 \), so the expression becomes \( 3x^{-3/2} \).

Step 3: Combine into Power Form

The expression \( 3x^{-3/2} \) is already in the power form \( ax^b \) where \( a = 3 \) and \( b = -3/2 \). Thus, the expression in power form is \( 3x^{-3/2} \).

Key Concepts

Simplifying Expressions

When simplifying expressions, our goal is to make them as straightforward as possible. This can involve combining like terms, reducing fractions, or transforming complex parts into simpler forms. In this exercise, we simplified a fraction \[ \frac{24}{(2 \sqrt{x})^{3}} \]The first step was simplifying the denominator, which was a composite of a constant and a square root raised to a power. By breaking it down further, we reduced it to its most understandable form. Remember these tips for simplification:

• Identify and simplify complex components like fractions and roots.
• Keep an eye out for common factors to further reduce expressions.
• Use basic arithmetical operations, such as division or multiplication, to simplify coefficients.

These steps will help you solve problems with less difficulty and more accuracy.

Exponents Rules

Exponential rules can greatly simplify the process of working with powers and roots. Understanding these rules allows you to manipulate expressions easily.Consider the expression \( (2 \sqrt{x})^{3} \). We used the rule for products:\[(ab)^n = a^n \cdot b^n\]Using this, \((2 \sqrt{x})^{3}\) became \(2^3 \cdot (\sqrt{x})^3\).Here are other useful exponent rules to remember:

• \((x^m)^n = x^{m \cdot n}\)
• \(x^m \cdot x^n = x^{m+n}\)
• \(x^0 = 1\) for any non-zero \(x\)

Apply these rules to breakdown expressions into simpler forms, which can then be combined or further reduced.

Fractional Exponents

Fractional exponents provide a way to deal with roots and powers simultaneously. They follow the format \(x^{m/n}\), where \(n\) is the root and \(m\) is the power.For instance, \(\sqrt{x} = x^{1/2}\). Raising \(\sqrt{x}\) to the power of 3, as in our denominator, can be written as:\[(x^{1/2})^3 = x^{3/2}\]The exponent \(3/2\) indicates a square root of \(x\), cubed. Why use fractional exponents?

• They simplify expressions involving both powers and roots.
• Making complex root expressions easier to manipulate within equations.
• Change between exponential and radical formats easily.

Understanding and utilizing fractional exponents can make solving complex expressions less intimidating and more intuitive.