Problem 102 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ (-3 m n)^{4} $$

Step-by-Step Solution

Answer

The simplified form of the expression \( (-3mn)^4 \) is \( 81m^4n^4 \)

Step 1: Expand the Exponent

Distribute the power of 4 to each term inside the parenthetical expression. Using exponentiation rule, the fourth power applies to both the -3 and each of the variables. So, \[ (-3m^n)^4 = (-3)^4 * m^4 * n^4 \]

Step 2: Calculate the Numerical Exponent

Calculate the exponent of -3. Remember that any negative number to an even power is positive. So, \[ (-3)^4 = 81 \]

Step 3: Combine Results

Combine the results from step 1 and 2 into a final answer. So, the solution is \[ 81m^4n^4 \]

Key Concepts

Understanding Exponentiation

Exponentiation is a fundamental mathematical operation. It involves raising a number, known as the base, to a power, called the exponent. This operation is written as \( a^n \), where \( a \) is the base, and \( n \) is the exponent, indicating how many times the base is multiplied by itself.
For example, in the expression \((-3)^4\):

• The base is -3.
• The exponent is 4, meaning \(-3\) is multiplied by itself four times.

Notably, when a negative number is raised to an even power, the result is positive. So, \((-3)^4 = 81\). Understanding this rule is essential because it affects the sign of the outcome in exponentiation.

Algebraic Expressions Simplified

Algebraic expressions are combinations of numbers, variables, and operators. They represent a particular value or set of values and are foundational in various mathematical computations.
In the expression \((-3mn)^4\):

• \(-3\) is a numerical coefficient.
• \(m\) and \(n\) are variables.

When simplifying such expressions that involve exponentiation, distribute the power to each factor within the parentheses. This means each term is raised to the indicated power: \((-3)^4 \cdot m^4 \cdot n^4\).
By understanding how to distribute exponents correctly, you can simplify more complex algebraic expressions with confidence.

Polynomials and Their Operations

Polynomials are expressions consisting of variables and coefficients, structured as a sum of terms. Each term includes a variable raised to a power. Simplifying polynomials often involves operations such as addition, subtraction, and multiplication.
In our expression, \(81m^4n^4\), after simplification:

• The term \(81\) is a constant coefficient.
• \(m^4\) and \(n^4\) are variables raised to the fourth power.

This expression is a monomial, a polynomial with only one term. When simplifying polynomial expressions, it’s crucial to understand how to manipulate terms systematically. This knowledge allows for effective problem-solving and application in various mathematical contexts.