Problem 41 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(x^{2} y^{3}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^{10} y^{15} \).

Step 1: Apply the Power of a Product Rule

The power of a product rule states that when you have a product raised to a power, you can distribute the exponent to each factor in the product. Here, apply the rule as follows: \( (x^2 y^3)^5 = (x^2)^5 (y^3)^5 \).

Step 2: Apply the Power Rule to Each Term

The power rule states that when you raise a power to another power, you multiply the exponents. Apply this to each term obtained in Step 1: \( (x^2)^5 = x^{2 imes 5} = x^{10} \) and \( (y^3)^5 = y^{3 imes 5} = y^{15} \).

Step 3: Combine the Results

Now, combine the results from Step 2 to obtain the simplified expression: \( x^{10} y^{15} \).

Key Concepts

Power of a Product

In mathematics, the "Power of a Product" rule is pivotal when working with exponents. It helps in simplifying expressions where a product is raised to a power. This rule states that when you have two or more factors within parentheses raised by an exponent, you can distribute that exponent to each factor inside separately. For example, given \[ (x^2 y^3)^5 \]we apply the power to each factor within the parentheses individually. So, it becomes:

• \( (x^2)^5 \) is one part
• \( (y^3)^5 \) is another part

This distributive approach makes calculations simpler because you're breaking down complex expressions into more manageable parts. Remember, it's akin to multiplying the exponents by the same factor for each component inside the parentheses.

Simplifying Expressions

Simplifying expressions involves reducing them into their simplest form so they're easier to work with or understand. To simplify an expression like \( (x^2)^5 (y^3)^5 \), you apply the power rule: when raising a power to another power, multiply the exponents.For \( (x^2)^5 \), multiply the exponents, resulting in \( x^{10} \).For \( (y^3)^5 \), multiply the exponents, resulting in \( y^{15} \).After performing these calculations:

• \( x^2 \, \) raised to the 5th power, becomes \( x^{10} \).
• \( y^3 \, \) raised to the 5th power, becomes \( y^{15} \).

Combining these results, you achieve the simplest form: \( x^{10} y^{15} \). Breaking down each element and recombining them reduces complex expressions with ease.

Exponents

Exponents are a way to express repeated multiplication. Understanding their behavior is key to properly simplifying expressions. An exponent tells you how many times to multiply a base number by itself. In the expression \( x^2 \), the base is \( x \) and the exponent is 2, meaning \( x \times x \).When dealing with compounded expressions like \( (x^2)^5 \), you multiply the exponents rather than dealing with the base repeatedly.To handle:

• \( (x^2)^5 \), the computation simplifies to multiplying the exponents: \( 2 \times 5 = 10 \), so it becomes \( x^{10} \).
• \( (y^3)^5 \), use the same multiplication method: \( 3 \times 5 = 15, \) leading to \( y^{15} \).

Understanding exponents allows one to efficiently process and simplify expressions by making seemingly complex calculations straightforward and more approachable.