Problem 23 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( \frac{9y^6}{x^8} \).

Step 1: Apply the Power of a Power Rule

The expression is \(\left( \frac{1}{3} x^{4} y^{-3} \right)^{-2}\). According to the power of a power rule, \( (a^m)^n = a^{m \cdot n} \). Apply this rule to each component inside the parentheses: \( (\frac{1}{3})^{-2} \), \( (x^4)^{-2} \), and \( (y^{-3})^{-2} \).

Step 2: Simplify the Coefficient

For the coefficient, \( (\frac{1}{3})^{-2} \) is equivalent to \( (3^1)^{2} \). Therefore, \( (\frac{1}{3})^{-2} = 3^2 = 9 \).

Step 3: Simplify the Variable with Positive Exponent

For \( x^{4} \): \( (x^4)^{-2} = x^{4 \cdot (-2)} = x^{-8} \).

Step 4: Simplify the Variable with Negative Exponent

For \( y^{-3} \): \( (y^{-3})^{-2} = y^{-3 \cdot (-2)} = y^6 \). Applying the power of a power rule, multiplying two negatives makes the second power positive.

Step 5: Combine the Simplified Expressions

Combine all the simplified parts: \( 9 \cdot x^{-8} \cdot y^6 \).

Step 6: Rewrite with Positive Exponents

To express the term with a positive exponent, recall that \( a^{-m} = \frac{1}{a^m} \). Thus, \( x^{-8} = \frac{1}{x^8} \). The expression becomes \( \frac{9y^6}{x^8} \).

Key Concepts

Power of a Power Rule

The **Power of a Power Rule** is a fundamental aspect of exponentiation that simplifies expressions when an exponent is raised to another exponent. This rule states that if you have a power inside the parentheses and raise it to another power, you multiply the exponents together. Consider the mathematical expression such as \( (a^m)^n = a^{m \cdot n} \).
For instance, in the given exercise where you have \( (x^4)^{-2} \), you apply the power of a power rule by multiplying 4 and -2 to transform it into \( x^{-8} \). Similarly, with \( (y^{-3})^{-2} \), applying the rule turns it into \( y^{6} \) since the product of two negatives yields a positive, making the exponent positive.
Using this rule simplifies the calculation process considerably and is an essential technique in algebra for handling complex exponentiation expressions.

Negative Exponents

**Negative Exponents** can seem tricky at first, but they simply tell you that a number is being divided rather than multiplied. A negative exponent corresponds to the reciprocal of the base raised to the positive of that exponent. In general, \( a^{-m} = \frac{1}{a^m} \).
In the context of the exercise \((x^4)^{-2}\), the outcome is \(x^{-8}\), which further simplifies to \(\frac{1}{x^8}\), reflecting the reciprocal action implied by the negative exponent.
Similarly, when simplifying \( (y^{-3})^{-2}\), the double negative from the exponent multiplication changes it to a positive exponent, resulting in \(y^6\). Understanding negative exponents is key to simplifying expressions and rearranging them to clear, user-friendly forms.

Simplifying Expressions

**Simplifying Expressions** involves reducing them to their most basic form without changing their value, which is crucial for clearer communication in mathematics. After using exponent rules, like the power of a power and dealing with negative exponents, the next step is to combine and reduce terms.
In this exercise, after applying the power of a power rule and rewriting negative exponents, you arrive at \( 9 \cdot x^{-8} \cdot y^6 \). To further simplify, you convert \( x^{-8} \) to its reciprocal form \( \frac{1}{x^8} \), thus obtaining the simplified final expression: \( \frac{9y^6}{x^8} \).
The simplification process makes mathematical expressions easier to handle and interpret, which is especially helpful when dealing with large and complex equations.