Problem 30 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ \left(-2 m^{4} n^{6}\right)^{2} $$

Step-by-Step Solution

Answer

The final representation of the given expression as a product of powers is \(4 \cdot m^8 \cdot n^{12}\).

Step 1: Apply the power of a product rule

According to the power of a product rule, we can break up the power of the initial term into a product of the powers of its factors. So starting with \(\left(-2 m^{4} n^{6}\right)^{2}\), you have \((-2)^2 \cdot (m^4)^2 \cdot (n^6)^2\).

Step 2: Apply the power of a power rule

Now, using power of a power rule, each component within parenthesis can be raised to the 2nd power individually. That leads to \((-2)^2 = 4\), \((m^4)^2 = m^{4 \cdot 2} = m^8\), \((n^6)^2 = n^{6 \cdot 2} = n^{12}\). Loosely speaking, the exponents are multiplied by the outer exponent in each case.

Step 3: Write the final representation

Combine the results of step 2 together to create the final answer: \(4 \cdot m^8 \cdot n^{12}\)

Key Concepts

Power of a Product

The power of a product rule is an essential concept in exponentiation. This rule tells us how to handle an expression where a power is applied to a product of different bases. For example, consider the expression \( \left(-2 m^{4} n^{6}\right)^{2} \). Instead of raising the entire product directly to the power, the power of a product rule allows us to distribute that power to each factor within the expression individually. By distributing, we transform

• \( (-2)^{2} \)
• \( (m^4)^{2} \)
• \( (n^6)^{2} \)

This approach simplifies calculations by breaking the problem into smaller pieces. Knowing this rule saves time and reduces errors when working with complex algebraic expressions.

Power of a Power Rule

The power of a power rule is a vital part of simplifying expressions that involve repeated exponentiation. When you have an expression of the form \((b^n)^m\), the power of a power rule states that you multiply the exponents. This concept is applied when you see a base with an exponent that is further raised to another exponent. In the exercise example, we have

• \((m^4)^2\)
• \((n^6)^2\)

Applying the power of a power rule, we multiply the exponents:

• \(4 \times 2 = 8\), giving us \(m^8\)
• \(6 \times 2 = 12\), resulting in \(n^{12}\)

By effectively using this rule, complex exponentiation questions can be broken down into manageable steps. This rule is not just useful but necessary for dealing with higher-level algebraic expressions.

Simplifying Expressions

Once you've applied the appropriate rules for powers, simplifying the expression becomes crucial. Simplification in mathematics involves rewriting an expression as neatly and concisely as possible. After distributing the exponent and applying the power of a power rule, the expression from our exercise is transformed. Initially, we had

• \((-2)^2\)
• \(m^8\)
• \(n^{12}\)

Simplifying each component, we note

• \((-2)^2 = 4\)

This allows us to combine the components into a single, simplified expression: \(4 \cdot m^8 \cdot n^{12}\). By reducing expressions to their simplest form, calculations are easier to perform and the final output becomes cleaner. Simplification is not just about making expressions look nice, but also about facilitating easier mathematical operations moving forward.