Problem 59 - Brief Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{4}{\sqrt[3]{8 x^{4}}} $$

Step-by-Step Solution

Answer

The expression in power form is \( 2x^{-4/3} \).

Step 1: Simplify the Denominator

First, identify the cube root in the denominator: \( \sqrt[3]{8x^4} \). The cube root of \(8\) is \(2\), and to find the cube root of \(x^4\), you can express it as \((x^4)^{1/3}\). Thus, the denominator becomes \( 2x^{4/3} \).

Step 2: Write the Expression as a Fraction

Now that the denominator is simplified to \(2x^{4/3}\), the entire expression is \( \frac{4}{2x^{4/3}} \). This can be further simplified by dividing the numerator by the constant in the denominator: \( \frac{4}{2} = 2 \). Thus, the expression becomes \( \frac{2}{x^{4/3}} \).

Step 3: Apply Negative Exponent Rule

Using the negative exponent rule that \( \frac{1}{x^a} = x^{-a} \), rewrite \( \frac{2}{x^{4/3}} \) as \( 2x^{-4/3} \). Thus, the expression in power form is \( 2x^{-4/3} \).

Key Concepts

Simplifying Expressions

Simplifying expressions means making them easier to understand or work with. It often involves condensing complex fractions, like the one in our example: \( \frac{4}{\sqrt[3]{8x^4}} \), to a more straightforward form.

When simplifying, you'll often apply mathematical rules and properties such as:

• Combining Like Terms: Terms with the same variable and exponent are added together.
• Reducing Fractions: Numerical parts are divided to their simplest form, like reducing \(\frac{4}{2}\) to \(2\).
• Rewriting Roots as Exponents: For example, cube roots can be rewritten using fractional exponents which makes them easier to handle algebraically.

In our problem, after identifying and simplifying the cube root in the denominator, the fraction becomes \(\frac{2}{x^{4/3}}\). This expression is much simpler to work with for further mathematical operations.

Cube Roots

Cube roots are a way to determine what number must be multiplied by itself three times to produce a given value. Mathematically, the cube root of a number \(a\) is expressed as \(\sqrt[3]{a}\), which is the same as \(a^{1/3}\).

This method of expressing roots as fractional exponents is particularly useful for simplifying algebraic expressions.

For instance,

• The cube root of 8 is 2, since \(2 \times 2 \times 2 = 8\).
• Similarly, \(x^4\) as a cube root becomes \((x^4)^{1/3} = x^{4/3}\).

In our example, we used these principles to transform \(\sqrt[3]{8x^4}\) into \(2x^{4/3}\). Understanding this conversion from cube roots to fractional exponents is essential in algebra for simplifying and manipulating expressions.

Negative Exponents

The concept of negative exponents is quite straightforward once you know the rule: \(x^{-a} = \frac{1}{x^a}\). This means, a negative exponent signifies the reciprocal of the base raised to a positive exponent.

Negative exponents can be confusing, but if you remember that it essentially flips the base to the denominator (or vice versa if dealing with fractions initially), it becomes a useful tool.

For example:

• \(x^{-2} = \frac{1}{x^2}\).
• Switching \(\frac{1}{x^{4/3}}\) to \(x^{-4/3}\) simplifies the expression by eliminating the fraction.

In our original problem, this understanding transformed \(\frac{2}{x^{4/3}}\) into \(2x^{-4/3}\), which is a more streamlined power form expression. When dealing with complex expressions, utilizing negative exponents can facilitate easier manipulation and calculation.