Problem 60 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\left(\frac{3 x^{4}}{y}\right)^{-3}$$

Step-by-Step Solution

Answer

\(\frac{y^{3}}{27 x^{12}}\)

Step 1: Apply the rule of exponent to the expression

According to the power of a quotient rule, \(\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}\), so \(\left(\frac{3 x^{4}}{y}\right)^{-3}\) becomes \(\frac{(3x^{4})^{-3}}{y^{-3}}\)

Step 2: Simplify the expression further

For any non-zero number a, a to the power of negative n is equal to 1 divided by a to the power of n (\(a^{-n} = \frac{1}{a^n}\)) and apply the same rule to \(3x^{4}\) and \(y\) we get \( \frac{1}{(3x^{4})^{3}} * (y^3) = \frac{1}{3^3 x^{12}} * y^{3}\)

Step 3: Continue to simplify the expression

= \(\frac{y^{3}}{3^{3} x^{12}} = \frac{y^{3}}{27 x^{12}}\)

Key Concepts

Power of a Quotient Rule

The power of a quotient rule is a fundamental concept in algebra that deals with simplifying expressions where a fraction is raised to an exponent. To apply this rule, you take an expression of the form \(\left(\frac{a}{b}\right)^{n}\) and raise both the numerator and the denominator to the power of \(n\). This results in \(\frac{a^{n}}{b^{n}}\), where \(a\) and \(b\) can be any real numbers, and \(n\) is an integer. It's particularly useful when working with complex expressions as it allows you to simplify them step by step.

For instance, if we have the expression \(\left(\frac{3 x^{4}}{y}\right)^{-3}\), we first apply the power of a quotient rule to get \(\frac{(3x^{4})^{-3}}{y^{-3}}\). Here, both the numerator \(3x^{4}\) and the denominator \(y\) are raised to the power of \(–3\), simplifying the complex fraction in the exponent to a more manageable form before further simplification.

Negative Exponents

The concept of negative exponents represents another crucial element of algebra. When you encounter a negative exponent, it indicates that the base should be reciprocated and then raised to the absolute value of the exponent. Mathematically speaking, for any non-zero number \(a\), and a positive integer \(n\), \(a^{-n} = \frac{1}{a^n}\).

In the earlier example, after applying the power of a quotient rule, we have \(3x^{4}\) and \(y\) raised to the power of \(–3\). To simplify, we reciprocate and change the negative exponent to a positive one, resulting in \(\frac{1}{(3x^{4})^{3}} * y^{3}\). This step is pivotal because it transforms the expression into a form where we can multiply and reduce as needed, continuing the simplification process.

Exponents Properties

Exponents properties, also known as the laws of exponents, include a variety of rules that dictate how to handle calculations involving powers. Beyond the negative exponent rule, other properties such as the product of powers rule, the quotient of powers rule, and the power of a power rule can greatly assist in simplifying expressions.

Continuing from our previous steps, we see these properties in action when we simplify \(\frac{1}{(3x^{4})^{3}} * y^{3}\) to get \(\frac{y^{3}}{3^{3} x^{12}}\), which then simplifies to \(\frac{y^{3}}{27 x^{12}}\). It is important to apply these properties correctly to achieve the simplest form. For instance, \(x^{a} * x^{b} = x^{a+b}\) or \(\frac{x^{a}}{x^{b}} = x^{a-b}\). Understanding these properties not only simplifies calculations but also ensures accuracy in algebraic manipulations.