Problem 48 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(\frac{x y}{7}\right)^{2} $$

Step-by-Step Solution

Answer

\(\frac{x^{2} y^{2}}{49}\)

Step 1: Understand the Expression

We start with the expression \( \left(\frac{x y}{7}\right)^{2} \). This is a power of a quotient, and we will apply the power rule to simplify it.

Step 2: Apply the Power of a Quotient Rule

The power of a quotient rule states that \( \left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}} \). Using this rule, we simplify our expression: \( \left(\frac{x y}{7}\right)^{2} = \frac{(x y)^{2}}{7^{2}} \).

Step 3: Apply the Power of a Product Rule

Next, apply the power of a product rule to the numerator. The rule is \( (ab)^{n} = a^{n} b^{n} \). Thus, \( (x y)^{2} = x^{2} y^{2} \).

Step 4: Simplify the Expression

Now, substitute back to get \( \frac{x^{2} y^{2}}{49} \). There are no further simplifications needed.

Key Concepts

Power Rule

The Power Rule is a basic yet crucial concept in algebra that helps us simplify expressions involving exponents. In essence, the Power Rule allows us to manage the powers when we have a power raised to another power. In simpler terms, if you have something like

• \((a^n)^m\), using the Power Rule, you simplify it to \(a^{n imes m}\).

Think of the power rule as a way to multiply exponents. It's especially useful when dealing with complex expressions containing multiple layers of exponents.
In our original expression, we apply this rule indirectly after breaking down the components using product and quotient rules. When expressions are nested, keep in mind the chain of operations that allows you to apply the Power Rule effectively among other strategies.

Power of a Quotient Rule

The Power of a Quotient Rule is specifically for situations where you have a fraction raised to a power. This rule tells us how to deal with these situations:
The rule states that

• \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\)

This means you can distribute the exponent across both the numerator and the denominator.
In the given exercise, we first encountered the expression \(\left(\frac{xy}{7}\right)^2\). Applying this rule, we broke it down to \(\frac{(xy)^2}{7^2}\). Each part of the fraction now has its own power, making it easier to simplify further.
This rule is very handy when simplifying expressions that start as fractions with complex numerators or denominators.

Power of a Product Rule

The Power of a Product Rule helps us deal with exponents applied to products of numbers or variables. The rule is straightforward:

• \((ab)^n = a^n b^n\)

This means that if you have two or more multiplied terms raised to a power, you can apply the power to each term separately.
In the example expression, after applying the power of a quotient rule, we have the term \((xy)^2\) in the numerator. Utilizing the Power of a Product Rule, we then simplify this to \(x^2 y^2\).
Applying these rules step-by-step turns complex expressions into simpler forms, greatly aiding in understanding and solving algebraic problems.