Problem 22 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(-6 m^{4}\right)^{2}$$

Step-by-Step Solution

Answer

The short version of the answer is: \(\left(-6 m^{4}\right)^{2} = 36m^8\)

Step 1: Apply the exponent rule to each component of the expression

We will apply the exponent \(2\) to both the constant, \(-6\), and the variable, \(m^4\). The exponent rule is: \((ab)^n = a^n \cdot b^n\). \( \left(-6 m^{4}\right)^{2} = (-6)^2 (m^4)^2 \)

Step 2: Simplify the constant term

Now we'll simplify \((-6)^2\) by multiplying \(-6\) by itself. \((-6)^2 = 36\)

Step 3: Apply the exponent rule to the variable term

To simplify the variable term \((m^4)^2\), we need to use the exponent rule \(a^{mn} = (a^m)^n\). In this case, our base is \(m\), and we're multiplying the exponents \(4\) and \(2\). \((m^4)^2 = m^{4 \times 2} = m^8\)

Step 4: Combine the simplified terms

Now that we have simplified both the constant and variable terms, we'll combine them. \(36 \cdot m^8 = 36m^8\) So, the simplified expression is: \( \left(-6 m^{4}\right)^{2} = 36m^8 \)

Key Concepts

Exponent Rules

When dealing with expressions that contain exponents, it's important to understand and apply the rules of exponentiation correctly. These rules help simplify expressions and make calculations more efficient. Here are some fundamental exponent rules you should know:

• Product of Powers: When multiplying two exponents with the same base, add the exponents, i.e., \( a^m imes a^n = a^{m+n} \).
• Power of a Power: When taking an exponent to another exponent, multiply the exponents, i.e., \( (a^m)^n = a^{m imes n} \).
• Power of a Product: Apply the exponent to each factor within the parentheses, i.e., \( (ab)^n = a^n imes b^n \).
• Zero Exponent: Any nonzero base raised to the zero power equals one, i.e., \( a^0 = 1 \).
• Negative Exponent: A negative exponent indicates a reciprocal, i.e., \( a^{-n} = \frac{1}{a^n} \).

Understanding these rules makes it simpler to handle complex algebraic expressions. In the given problem, the power of a product rule was applied to dissect and simplify the expression \( (-6m^4)^2 \). This allowed us to separately deal with the constant and the variables, illustrating the power of these fundamental rules for managing expressions.

Simplification of Expressions

Simplifying expressions means reducing them to their most basic form. This often involves making use of exponent rules, like combining similar terms and removing parentheses if possible. A clear simplification is invaluable in making expressions easier to work with.

Let's take a closer look at the process of simplifying \( (-6m^4)^2 \):

• Apply the Exponent: We start by separately applying the squared exponent to both the number -6 and the variable \( m^4 \). This step gives us the components \( (-6)^2 \) and \( (m^4)^2 \).
• Simplify the Constant: The calculation for the number \( (-6)^2 \) is straightforward: \( (-6) imes (-6) = 36 \), removing the negative sign.
• Simplify the Variable: For the variable part, we multiply the exponents as per the power of a power rule, resulting in \( m^{4 imes 2} = m^8 \).
• Combine Results: Finally, we combine the simplified parts to achieve the clean form: \( 36m^8 \).

This series of steps demonstrates how breaking down complex terms into smaller, simpler parts makes it easier to see the overall picture.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and exponentiation). Simplifying algebraic expressions allows you to work with them more effectively, enhancing both understanding and problem-solving efficiency.

In \( (-6m^4)^2 \), you're only dealing with multiplication raised to a power and variables. Here's what's happening with this expression:

• Constant and Variables: An algebraic expression can contain constants such as numbers, and variables which are symbols representing unknown values. In our case, \( -6 \) is a constant, and \( m^4 \) is a variable raised to a power.
• Operation of Powers: Raising the expression \( (-6m^4) \) to the power of 2 involves using the exponent rules discussed. The main operation here is squaring, which intensifies the base numbers and variables.
• Resultant Simplification: Simplifying the terms using exponent rules leads to \( 36m^8 \), which is the simplified form of the original expression, showcasing the result of this algebraic manipulation.

Understanding algebraic expressions' components and operations is crucial for successfully managing such mathematical statements, leading to a firm grasp of algebra and its practical applications.