Problem 45 - Precalculus: Functions and Graphs

Question

Simplify. $$\frac{\left(x^{6} y^{3}\right)^{-1 / 3}}{\left(x^{4} y^{2}\right)^{-1 / 2}}$$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Apply Exponent Rule for Negative Exponents

Recall that \( (a^m)^{-n} = (a^{-m})^n \). Start by applying this rule to both the numerator and the denominator to get rid of the negative exponents: - Numerator: \( \left(x^6 y^3\right)^{-1/3} = \left(x^{-6} y^{-3}\right)^{1/3} = x^{-2} y^{-1} \) - Denominator: \( \left(x^4 y^2\right)^{-1/2} = \left(x^{-4} y^{-2}\right)^{1/2} = x^{-2} y^{-1} \).

Step 2: Simplify Both Terms

Now, simplify both terms: - The numerator simplifies to \( x^{-2} y^{-1} \) - The denominator simplifies to \( x^{-2} y^{-1} \)

Step 3: Simplify the Fraction

Since both the numerator and the denominator are the same, \( \frac{x^{-2} y^{-1}}{x^{-2} y^{-1}} = 1 \).

Step 4: Final Result

Rationalizing the expression, we find that the simplified form is simply \( 1 \), since any number divided by itself equals \( 1 \).

Key Concepts

Negative Exponents

When dealing with exponents, you might encounter a negative exponent. This can seem tricky at first, but it really just means a reciprocal. Generally, when you see an exponent written like this: \( x^{-a} \), it can be thought of as the reciprocal of \( x^a \). In simple terms, you can rewrite it as \( \frac{1}{x^a} \).
This transformation holds true for any expression with a negative exponent. For example, if you begin with \( (x^6 y^3)^{-1/3} \), you rewrite it by finding the reciprocal so that it becomes \( \left( \frac{1}{x^6 y^3} \right)^{1/3} \). This allows you to eliminate the negative sign within the exponent, making further calculations more straightforward.

Numerator and Denominator Simplification

This process involves changing more complex expressions, aimed at making operations easier. In the original problem, after applying the negative exponent rules, we arrived at \( x^{-2}y^{-1} \) for both the numerator and the denominator.

Each component in these expressions can be seen as a fraction in disguise. For instance, \( x^{-2}y^{-1} \) is essentially \( \frac{1}{x^2y} \). Applying this thought process to both numerator and denominator uniformly simplifies upcoming steps because it often reveals whether the overall fraction can become simpler or even potentially evaluate to a simple number.

Fraction Simplification

Simplifying fractions means reducing them to their simplest form. This involves dividing the numerator and the denominator by their greatest common factor. In our context, both the numerator and the denominator turned into \( x^{-2}y^{-1} \).

• Since both parts of the fraction are identical, dividing them results in 1 because any number divided by itself equals 1.
• This is a fundamental step in simplifying any rational expression and is key to reaching the final simplest form of the equation.

With the example studied, think of simplification like finding out how many times one expression fits into the other, and here, they go perfectly once. Easy, right?

Rationalization of Expressions

Rationalizing expressions involves transforming them to avoid negative exponents or assure a simpler form. The primary aim is to ensure that any radicals or expressions are in the most universally understood form.
In this exercise, when both the numerator and the denominator were identical and fully simplified, the result was 1. This step ensures clarity in further mathematical operations or when dealing with complex numbers.
In broader terms, rationalization helps us work with numbers or expressions that are simpler and clearer, enhancing ease of calculation and reducing possible errors in continuing operations. Working towards a rationalized form is all about making life simpler, mathematically speaking!