Problem 48 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponent(s). $$ \left(\frac{x y^{-2} z^{-3}}{x^{2} y^{3} z^{-4}}\right)^{-3} $$

Step-by-Step Solution

Answer

\( \frac{x^{3} y^{15}}{z^{3}} \) is the simplified expression.

Step 1: Apply the Negative Exponent Rule

First, apply the negative exponent rule to the entire expression. The negative exponent outside the parentheses means we take the reciprocal of the inside expression and raise it to the positive of the exponent. Thus, \[ \left( \frac{x y^{-2} z^{-3}}{x^{2} y^{3} z^{-4}} \right)^{-3} = \left( \frac{x^{2} y^{3} z^{-4}}{x y^{-2} z^{-3}} \right)^{3}.\]

Step 2: Simplify Inside the Parentheses

Before applying the power, simplify the fraction inside the parentheses by dividing powers of the same base: \[ \frac{x^{2} y^{3} z^{-4}}{x y^{-2} z^{-3}}.\] This becomes \[ \frac{x^{2-1} y^{3-(-2)} z^{-4-(-3)}} = x^{1} y^{5} z^{-1}.\]

Step 3: Apply the Power to the Simplified Expression

Now that we have \( x^{1} y^{5} z^{-1} \), apply the exponent of 3: \[ \left(x^{1} y^{5} z^{-1}\right)^{3} = x^{3} y^{15} z^{-3}.\]

Step 4: Eliminate Negative Exponents

Address the negative exponent in \( z^{-3} \) by moving it to the denominator: \[ x^{3} y^{15} \cdot \frac{1}{z^{3}}.\] Thus, we get \[ \frac{x^{3} y^{15}}{z^{3}}.\]

Key Concepts

Negative Exponents

When you see a negative exponent, it can seem a bit intimidating. However, it's simply a way to express reciprocals. If you have an expression like \( a^{-n} \), it equals \( \frac{1}{a^{n}} \). This means that you take the reciprocal of the base and change the sign of the exponent to positive.
This concept is handy when dealing with fractions raised to negative exponents. For example, in the problem we started with: * The expression \( \left( \frac{x y^{-2} z^{-3}}{x^{2} y^{3} z^{-4}} \right)^{-3} \) * This involves turning the whole fraction upside-down and changing the sign of the exponent to positive, making it easier to simplify in later steps.

Simplifying Fractions

Simplifying fractions is all about reducing the fraction to its simplest form. It involves subtracting or adding the exponents, depending upon whether you're multiplying or dividing them.
In this exercise, to simplify \( \frac{x^{2} y^{3} z^{-4}}{x y^{-2} z^{-3}} \), we:

• Subtract the exponent of \( x \): \( x^{2-1} = x^{1} \)
• Adjust the exponents of \( y \): \( y^{3-(-2)} = y^{5} \)
• Change the exponent of \( z \): \( z^{-4-(-3)} = z^{-1} \)

This process of subtracting exponents is what simplifies the expression inside the parentheses initially.

Power of a Quotient

The power of a quotient rule states that when you raise a fraction to an exponent, you apply that exponent to both the numerator and the denominator. This rule makes handling complex expressions more straightforward.
For the exercise expression \( \left(x^{1} y^{5} z^{-1}\right)^{3} \), applying the power, we get:

• \(x^{1 \times 3} = x^{3}\)
• \(y^{5 \times 3} = y^{15}\)
• \(z^{-1 \times 3} = z^{-3}\)

By applying the exponent to each factor inside the parentheses, we maintain the integrity of the expression while rendering it easier to handle.

Reciprocal Rule

The reciprocal rule is a crucial concept when dealing with negative exponents and simplifying expressions. In simple terms, the reciprocal of a number is flipping it upside down. For instance, the reciprocal of \( z^{-3} \) is \( \frac{1}{z^{3}} \).
This is particularly important when eliminating negative exponents from an expression. When encountering \( z^{-3} \) as in our solution, you move \( z^{3} \) to the denominator to make the exponent positive, resulting in \( \frac{1}{z^{3}} \). This technique clears negative exponents from the expression, simplifying it to its final form: \( \frac{x^{3} y^{15}}{z^{3}} \).