Problem 55 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(\frac{x^{-1}}{y^{-4}}\right)^{-3}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x^3}{y^{12}}\).

Step 1: Apply the Power of a Quotient Rule

Recall that when you raise a quotient to a power, you apply the exponent to both the numerator and denominator separately. The expression \(\left(\frac{x^{-1}}{y^{-4}}\right)^{-3}\) can be rewritten as \(\frac{(x^{-1})^{-3}}{(y^{-4})^{-3}}\).

Step 2: Simplify Exponents within the Fraction

Using the rule \((a^m)^n = a^{m \cdot n}\), simplify both the numerator and the denominator. For the numerator, \((x^{-1})^{-3} = x^{3}\). For the denominator, \((y^{-4})^{-3} = y^{12}\). This gives us \(\frac{x^{3}}{y^{12}}\).

Step 3: Express without Negative Exponents

Since the expression \(\frac{x^{3}}{y^{12}}\) already has positive exponents and no zero exponents, no further simplification regarding the exponent signs is needed.

Key Concepts

Power of a Quotient Rule

When dealing with exponents, the Power of a Quotient Rule is pivotal for simplifying expressions. This rule states that when you have a quotient raised to an exponent, you need to apply that exponent to both the numerator and the denominator separately. For example, given \[\left(\frac{x^{-1}}{y^{-4}}\right)^{-3},\]you apply the -3 exponent to both \( x^{-1} \) and \( y^{-4} \).
This results in:\[\frac{(x^{-1})^{-3}}{(y^{-4})^{-3}}.\]
This step is crucial because it breaks down a complex expression into parts that are much easier to manage.
Remember, when you apply the Power of a Quotient rule, you're preparing your expression for further simplification by distributing the exponent correctly across both the parts of the fraction.

Simplifying Exponents

Once you have applied the Power of a Quotient rule, the next step is simplifying the exponents of both the numerator and the denominator. The law of exponents \((a^m)^n = a^{m \cdot n}\) allows us to simplify expressions where an exponent is raised to another exponent.
For example, after applying the Power of a Quotient Rule in\[\frac{(x^{-1})^{-3}}{(y^{-4})^{-3}},\]
we simplify the numerator \((x^{-1})^{-3}\) into \(x^{3}\), because \(-1 \times -3 = 3\).
And for the denominator \((y^{-4})^{-3}\), it becomes \(y^{12}\), since \(-4 \times -3 = 12\).
Keep in mind, simplifying both parts into positive exponents is a step towards expressing the final answer without negative or zero exponents.

Negative Exponents

Negative exponents can initially appear intimidating, but understanding them simplifies the process significantly. A negative exponent means that you take the reciprocal of the base. For \(a^{-n}\), it is equivalent to \(\frac{1}{a^n}\).
When given an expression like\[\left(\frac{x^{-1}}{y^{-4}}\right)^{-3},\]
you encounter negative exponents multiple times. When simplifying, as done in the original steps, turning \(x^{-1}\) into \(x^{3}\) through multiplication \(-1 \times -3 = 3\) turns a reciprocal back into a straightforward exponentiation.
Once all exponents are positive as in the final result \(\frac{x^{3}}{y^{12}}\),you’ve successfully applied your understanding of negative exponents to rewrite the expression effectively. Always aim for positive exponents in the final expression for clarity and simplicity.