Problem 62 - 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: Simplify the Denominator

The given expression is \( \frac{18}{(3 \sqrt[3]{x})^{2}} \). Start by simplifying the denominator. The term \( 3 \sqrt[3]{x} \) can be rewritten as \( 3 (x^{1/3}) \). Thus, raise it to the power of 2: \((3 (x^{1/3}))^2 = 3^2 \cdot (x^{1/3})^2 = 9x^{2/3}\). The expression now turns into \( \frac{18}{9x^{2/3}} \).

Step 2: Divide and Simplify

Divide the numerator by the simplified denominator. The fraction \( \frac{18}{9x^{2/3}} \) can be rewritten by simplifying the constant terms: \( \frac{18}{9} = 2 \). Therefore, \( \frac{18}{9x^{2/3}} = \frac{2}{x^{2/3}} \). This simplifies to an expression of the form \( 2x^{-2/3} \) because dividing by \( x^{2/3} \) is equivalent to multiplying by \( x^{-2/3} \).

Key Concepts

Simplifying Expressions

Simplifying expressions can seem daunting at first, but it boils down to reducing them to their simplest form. This process involves manipulating algebraic expressions by combining like terms, factoring, or applying mathematical operations to rewrite them in a clearer way.

In our exercise, the simplification began by looking at a complex fraction, \( \frac{18}{(3 \sqrt[3]{x})^{2}} \). The goal is to make calculations easier and expressions more manageable. Start simplification by tackling the denominator: expressing \( 3 \sqrt[3]{x} \) as \( 3x^{1/3} \).

This turns the denominator into \((3x^{1/3})^2\), which when expanded gives \(9x^{2/3}\). Simplifying further, you divide \(18\) by \(9x^{2/3}\), simplifying \(\frac{18}{9} = 2\). As a result, you transform the whole expression into a more digestible form \(2x^{-2/3}\).

This ease of manipulation and calculation principles apply not only to fractions but many expressions you'll encounter.

Power Form

The power form is a way of expressing numbers or variables using exponents. It is useful for simplifying calculations and expressing large numbers compactly. While this might seem complex, it's a key step in understanding algebra and manipulating mathematical equations.

In this context, power form means representing components of an expression using exponents. For instance, in \( 3x^{1/3} \), \(3\) is the base and \(x^{1/3}\) shows its power or exponent representation. Power form can either simplify complex expressions or make concise and clear expressions clearer.

It helps especially in the given exercise, where the original fraction \(\frac{18}{9x^{2/3}}\) was rewritten as \(2x^{-2/3}\). This transformation involved standard rules of exponents, where dividing by a fraction results in a negative exponent. Power form simplifies management of terms, making calculations straightforward and understandable.

Fractional Exponents

Fractional exponents are another way to represent roots. Instead of using radical symbols, we use exponents that are fractions. This conversion simplifies many algebraic operations and offers a unified way of handling both multiplication and root extraction.

Here, the cube root of \(x\), written as \(\sqrt[3]{x}\), is expressed as \(x^{1/3}\) using fractional exponents. This helps transition easily between multiplication and division, as what you see in the exercise when \( (x^{1/3})^2 \) becomes \(x^{2/3}\).

The application of fractional exponents also simplifies the expression \(\frac{2}{x^{2/3}}\), turning it into \(2x^{-2/3}\). Understanding fractional exponents helps with algebraic manipulations, notably when dealing with roots and polynomials. Mastering this concept provides an edge in simplifying and solving complex expressions.