Problem 93 - Intermediate Algebra

Question

Simplify each exponential expression. $$ \left(-2 x^{3} y^{2}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified expression is \\(-32x^{15}y^{10}\\).

Step 1: Apply the Power of a Product Rule

The power of a product rule states that \( (ab)^n = a^n imes b^n \). Apply this rule to the expression \( (-2x^3y^2)^5 \) by distributing the power of 5 to each component within the parentheses: \((-2)^5 imes (x^3)^5 imes (y^2)^5\).

Step 2: Simplify \\((-2)^5\\)

Calculate \( (-2)^5 \). Since \( 5 \) is an odd number, the result will be negative: \((-2)^5 = -32 \).

Step 3: Simplify \\((x^3)^5\\) using Power of a Power Rule

The power of a power rule states that \( (a^m)^n = a^{m imes n} \). Simplify \((x^3)^5\) as follows: \( (x^3)^5 = x^{3 imes 5} = x^{15} \).

Step 4: Simplify \\((y^2)^5\\) using Power of a Power Rule

Apply the power of a power rule to simplify \((y^2)^5\): \( (y^2)^5 = y^{2 imes 5} = y^{10} \).

Step 5: Combine the Simplified Components

Combine the results from Steps 2, 3, and 4: \(-32 imes x^{15} imes y^{10}\). Write the expression as \(-32x^{15}y^{10}\).

Key Concepts

Power of a Product Rule

When you encounter an exponential expression where the entire expression contained within parentheses is raised to an exponent, the power of a product rule comes handy. This rule tells us that if you have an expression like \((ab)^n\), you can distribute the exponent to both the base \(a\) and \(b\) within the parentheses. This means you rewrite the expression into the form \(a^n \times b^n\).

This rule simplifies working with products that are raised to a power by allowing us to separate and simplify each component individually. For example, in the exercise

• Expression: \((-2x^3y^2)^5\)
• Apply the power of a product: \((-2)^5 \times (x^3)^5 \times (y^2)^5\)

Each part of \((-2x^3y^2)\) is individually raised to the power of 5, allowing for simpler calculations.

Power of a Power Rule

The power of a power rule is crucial when dealing with expressions where a power is raised to another power. This rule states that \((a^m)^n = a^{m \times n}\). Essentially, you multiply the exponents. This can simplify complex exponential expressions by condensing them into a single power.

Applying this rule to the expression from the exercise:

• For \((x^3)^5\), apply the power of a power rule: \((x^3)^5 = x^{3 \times 5} = x^{15}\)
• Similarly, for \((y^2)^5\), substituting gives: \((y^2)^5 = y^{2 \times 5} = y^{10}\)

This step helps in reducing the number of operations by combining the powers into a single exponent, making it not only efficient but also easier to comprehend.

Simplifying Expressions

Simplifying expressions involves evaluating and rewriting them in their most concise form without changing their value. After applying the power rules, the next step is to calculate the powers and multiply the simplified parts together. Let's see this in action using our exercise as a guide.

The expression \((-2x^3y^2)^5\) has been broken down. Now each component is calculated:

• Simplifying \((-2)^5\): Since 5 is odd, the result is negative. So, \((-2)^5 = -32\).
• The simplified expression for \((x^3)^5\) is \(x^{15}\).
• The simplified expression for \((y^2)^5\) is \(y^{10}\).

Finally, combine all these parts:

• Combine: \(-32 \times x^{15} \times y^{10}\)
• Result: \(-32x^{15}y^{10}\)

Simplification ensures that an expression is as compact as possible, making it easier to understand and use in subsequent calculations or operations.