Problem 45 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(\frac{r}{s}\right)^{9} $$

Step-by-Step Solution

Answer

\( \frac{r^9}{s^9} \)

Step 1: Identify the expression

We have \( \left(\frac{r}{s}\right)^9 \) which is a quotient raised to a power. Our goal is to simplify this expression using the power of a quotient rule.

Step 2: Apply the Power of a Quotient Rule

According to the power of a quotient rule, if you have a quotient raised to a power \( \left( \frac{a}{b} \right)^n \), it can be expressed as \( \frac{a^n}{b^n} \). For this problem, apply the rule: \( \left( \frac{r}{s} \right)^9 = \frac{r^9}{s^9} \).

Step 3: Simplified Expression

After applying the power of a quotient rule, the expression \( \left( \frac{r}{s} \right)^9 \) simplifies to \( \frac{r^9}{s^9} \). This is the simplified form of the original expression.

Key Concepts

Power of a Product

When you encounter an expression where a product is raised to a power, such as \((ab)^n\), the power of a product rule is your best friend. This rule states that each factor within the parentheses should be raised to the power separately. Let's break it down:

• If you have \((ab)^n\), it becomes \(a^n \times b^n\).
• Think of it as sharing the exponent with each part inside the parentheses.

For example, with \((xy)^3\), you would turn this into \(x^3 \times y^3\). This simple rule makes it easier to manage expressions by distributing the power to individual factors for easier computation.
So when you see a product raised to a power, apply this rule to simplify your math life!

Power of a Quotient

Simplifying a quotient raised to a power works similarly to simplifying a product. When you have an expression in the form of \(\left(\frac{a}{b}\right)^n\), you apply the power of a quotient rule. This rule allows you to raise both the numerator and the denominator to the power separately.

• Use \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) to simplify.
• Each part of the fraction gets the exponent individually.

Consider the expression \(\left(\frac{r}{s}\right)^9\) from our exercise. We apply the rule to get \(\frac{r^9}{s^9}\).
This way, you maintain the ratio but express it in a simplified form, making computations more approachable and less intimidating.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest forms. By applying rules like the power of a product and the power of a quotient, you can make expressions more understandable and easy to work with.

• First, identify any powers or exponents in the expression.
• Next, apply relevant rules to distribute the exponents correctly.
• Finally, check if anything can be further reduced or combined.

In our exercise, we started with \(\left(\frac{r}{s}\right)^9\) and simplified it to \(\frac{r^9}{s^9}\), making it clear and manageable.
These steps not only streamline problems but enhance your ability to solve more complex equations by stripping them down to the basics. Simplification is an essential skill that makes math more accessible!