Problem 61 - Brief 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, we simplify the denominator \((2 \sqrt{x})^3\). Recognize that \(\sqrt{x} = x^{1/2}\), so \(2\sqrt{x}\) can be rewritten as \(2x^{1/2}\). Therefore, \((2\sqrt{x})^3 = (2x^{1/2})^3\).

Step 2: Expand the Exponent

We know that \((ab)^n = a^n \cdot b^n\). Apply this to \((2x^{1/2})^3 = 2^3 \cdot (x^{1/2})^3\). Calculate these separately: \(2^3 = 8\) and \((x^{1/2})^3 = x^{3/2}\). Combine to get: \(8x^{3/2}\).

Step 3: Express the Original Expression

Now write the original expression \(\frac{24}{(2 \sqrt{x})^{3}}\) as \(\frac{24}{8x^{3/2}}\). Simplify this by dividing both the numerator and the denominator by 8, resulting in \(\frac{3}{x^{3/2}}\).

Step 4: Convert to Power Form

The expression \(\frac{3}{x^{3/2}}\) can be rewritten as \(3x^{-3/2}\) by recognizing \(\frac{1}{x^{n}} = x^{-n}\). Thus, the expression in power form is \(3x^{-3/2}\).

Key Concepts

Exponents

Exponents are a shorthand way of expressing repeated multiplication of a number by itself. The expression of a number, say \(a\), raised to a power \(n\) is written as \(a^n\). This indicates that \(a\) is multiplied by itself \(n\) times. Understanding exponents is crucial in algebra because they simplify complex expressions and calculations.

Let's look at some examples:

• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(5^2 = 5 \times 5 = 25\)
• \(x^4 = x \times x \times x \times x\)

Exponents follow specific rules that make them useful, such as:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^n = a^n \cdot b^n\)

These rules help simplify expressions, especially when dealing with complex algebraic problems like simplifying \((2x^{1/2})^3\).

Simplifying Expressions

Simplifying expressions means transforming them into a more basic or digestible form while still having the same value. One of the methods frequently used in simplification is applying the rules of exponents.

For the example in our original exercise, simplification involved several steps:
First, we needed to simplify the denominator \((2\sqrt{x})^3\), rewritten as \((2x^{1/2})^3\) after identifying \(\sqrt{x}\) as \(x^{1/2}\). Next, applying the Power of a Product rule, we expanded this expression into \(2^3 \cdot (x^{1/2})^3\), which computes to \(8x^{3/2}\).

An essential aspect of simplification is realizing that the rules provide a systematic method to break down expressions into elementary parts. Dividing and reducing fractions, as we did with \(\frac{24}{8x^{3/2}}\) to \(\frac{3}{x^{3/2}}\), is also part of simplification, bringing expressions into their lowest terms for easier computation and understanding.

Fractional Exponents

Fractional exponents represent roots, and understanding them is key to manipulating algebraic expressions in power form. The notation \(x^{m/n}\) signifies the \(n\)-th root of \(x\) raised to the \(m\)-th power. For example, \(x^{1/2}\) means the square root of \(x\), while \(x^{3/2}\) indicates that you first take the square root of \(x\) and then cube the result.

In algebra, fractional exponents provide a more unified and powerful method to express roots alongside integer exponents. They allow you to simplify expressions like radicals, making calculations more manageable and systematic.

Returning to our exercise, we converted \(\frac{3}{x^{3/2}}\) to the power form \(3x^{-3/2}\) by applying the rule that \(\frac{1}{x^n} = x^{-n}\). This conversion to negative exponents shows how flexible and valuable fractional exponents are, enabling easier handling of division or root operations in power form.