Problem 62 - Brief Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \frac{18}{(3 \sqrt[3]{x})^{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Simplifying the Denominator

The expression is given as \( \frac{18}{(3 \sqrt[3]{x})^{2}} \). Begin by raising both the base \(3\) and \(\sqrt[3]{x}\) to the power of 2 inside the parentheses. This becomes: \( (3^2 (\sqrt[3]{x})^2) \). Recall that \(3^2 = 9\) and write \((\sqrt[3]{x})^2\) in radical form as \(x^{2/3}\). Thus, the denominator becomes \(9x^{2/3}\).

Step 2: Reducing the Fraction

Now, substitute the simplified form of the denominator back into the expression: \[ \frac{18}{9x^{2/3}} \].Divide the constants in the numerator and the denominator. Since \( \frac{18}{9} = 2 \), the expression simplifies to: \[ \frac{2}{x^{2/3}} \].

Step 3: Writing the Expression in Power Form

Convert the expression \( \frac{2}{x^{2/3}} \) into power form. Recall the exponent rule \( \frac{1}{x^b} = x^{-b} \).Applying this rule, we write the expression as:\[ 2x^{-2/3} \].Thus, the expression in the form \( ax^b \) is \( 2x^{-2/3} \).

Key Concepts

Power Form

Power form is a way to express mathematical expressions using exponents. This form is beneficial as it simplifies complex expressions and makes them easier to manipulate. When expressions are in the form of \(ax^b\), \(a\) represents a coefficient, and \(b\) is the exponent that shows how many times \(x\) is multiplied by itself.
The original exercise asked to write the expression in power form, transforming it into \(2x^{-2/3}\). Here:

• \(2\) is the coefficient, meaning it multiplies the expression.
• \(x^{-2/3}\) shows the power form of \(x\) including the exponent \(-2/3\), indicating an inverse cube root squared.

Power form provides a streamlined notation, especially useful in calculus and higher-level algebra.

Exponent Rules

Exponent rules help us manage and simplify expressions involving powers. These rules apply whether you multiply, divide, or raise powers to powers. Understanding these can quickly transform complex expressions into simpler forms.
Key rules include:

• Product of Powers: \(x^a \times x^b = x^{a+b}\)
• Quotient of Powers: \(\frac{x^a}{x^b} = x^{a-b}\)
• Power of a Power: \((x^a)^b = x^{a \cdot b}\)
• Negative Exponent: \(x^{-a} = \frac{1}{x^a}\)

In the original solution, the negative exponent rule was applied to express \(\frac{1}{x^{2/3}}\) as \(x^{-2/3}\). This transformation lets you write fractions as simpler power expressions. Exponent rules streamline working with powers and are foundational in simplifying algebraic expressions.

Simplifying Fractions

Simplifying fractions is a crucial step in algebra. It involves reducing the numerator and the denominator to their smallest value without changing the fraction's value.
For example, in the given exercise:

• Simplify the Constants: \(\frac{18}{9}\) turns into \(2\) because both numbers divide evenly by 9.
• Dividing Variables: With terms containing variables, apply exponent rules to simplify further.

In the exercise's solution, we see a fraction \(\frac{18}{9x^{2/3}}\) reduced to \(\frac{2}{x^{2/3}}\) by dividing the constants.
This form is easier to convert into power form. The simplified expression helps in future operations, making further algebraic manipulations more manageable. Simplifying fractions is essential for cleanly presenting solutions in algebraic contexts.