Problem 66 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \frac{(r s)^{-3}}{\left(r^{2} s^{3}\right)^{2}} $$

Step-by-Step Solution

Answer

\(\frac{1}{r^7 s^9}\)

Step 1: Apply Negative Exponent Rule

To simplify \((rs)^{-3}\), we use the rule that \(a^{-n} = \frac{1}{a^n}\). This converts \((rs)^{-3}\) to \(\frac{1}{(rs)^3}\).

Step 2: Expand the Power of a Product

For the term \((r^2 s^3)^2\), apply the rule \((a^m b^n)^p = a^{mp} b^{np}\). Thus, \((r^2 s^3)^2 = r^{4} s^{6}\).

Step 3: Substitute the Simplified Terms Back

Substitute the expressions from Steps 1 and 2 into the original equation to get: \[ \frac{1}{(rs)^3} \cdot \frac{1}{r^{4} s^{6}}. \]

Step 4: Combine the Denominator Terms

Combine the terms in the denominator by multiplying them: \((rs)^3 = r^3 s^3\), so the expression becomes \[ \frac{1}{r^3 s^3 r^4 s^6}. \]

Step 5: Simplify Using the Power Rule

Combine the powers of the same base in the denominator: \(r^3 r^4 = r^{3+4} = r^7\) and \(s^3 s^6 = s^{3+6} = s^9\). The expression simplifies to \[ \frac{1}{r^7 s^9}. \]

Key Concepts

Negative Exponent Rule

The Negative Exponent Rule is a key principle in algebra that allows us to transform terms with negative exponents into positive ones. According to this rule, whenever you have a term with a negative exponent, for example, \(a^{-n}\), it can be rewritten as the reciprocal of the term with a positive exponent: \(\frac{1}{a^n}\). This is a powerful tool for simplifying expressions.
In our example exercise, the expression \((rs)^{-3}\) was initially presented with a negative exponent. By applying the Negative Exponent Rule, we convert it to \(\frac{1}{(rs)^3}\). This step not only simplifies our expression but also sets the groundwork for further simplification processes.
Remember, when simplifying algebraic expressions, changing from negative to positive exponents is often necessary to reach a final, simplified form.

Power of a Product

The Power of a Product rule is essential for dealing with expressions raised to an exponent. This rule states that when a product \((a \cdot b)^n\) is raised to an exponent, the exponent applies to each factor inside the product: \(a^n \cdot b^n\).
In the provided exercise, we apply this rule to simplify \((r^2 s^3)^2\). By distributing the exponent 2 to both \(r^2\) and \(s^3\), we calculate:

• \(r^2\) to the power of 2 becomes \(r^{2 \times 2} = r^4\).
• Similarly, \(s^3\) to the power of 2 becomes \(s^{3 \times 2} = s^6\).

Thus, the expression \((r^2 s^3)^2\) simplifies to \(r^4 s^6\). Mastering this rule allows for efficient handling of terms within a product, ensuring each factor is correctly simplified.

Combining Powers

Combining Powers is another crucial concept, particularly when dealing with the multiplication of terms with the same bases. This rule tells us to add the exponents when multiplying similar bases: \(a^m \times a^n = a^{m+n}\).
When we simplified the expression \(\frac{1}{r^3 s^3 r^4 s^6}\) from our exercise, we applied this rule. We needed to combine the powers in the denominator. For the base \(r\), we had two exponents to combine:

• \(r^3\) and \(r^4\) combine to become \(r^{3+4} = r^7\).

Likewise, for the base \(s\):

• \(s^3\) and \(s^6\) combine to become \(s^{3+6} = s^9\).

Using the rule, we consolidate the expression to \(\frac{1}{r^7 s^9}\). This helps simplify the complex expression to its most compact and readable form.