Problem 28 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ (6 m n)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((6mn)^2\). Answer: $$36m^2n^2$$

Step 1: Apply the exponent to each part of the base

As given, \((6mn)^2\). To simplify this, we need to multiply the exponent by each part of the base. So, we will have: $$ (6^2)(m^2)(n^2) $$

Step 2: Compute for the constant number

In our previous step, we found that simplifying the expression will lead to \((6^2)(m^2)(n^2)\). Now we need to perform the square operation on the constant number, which is 6: $$ (36)(m^2)(n^2) $$

Step 3: Write the final simplified expression

Now that we have applied the exponent to each part of the base and simplified the constant number, we have our final simplified expression: $$ 36m^2n^2 $$

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex algebraic formulas into their simplest forms. This helps make calculations more manageable and reduces the chances of error. When using power rules for exponents, you need to distribute the exponent across all elements in parentheses.

This means for an expression like \((6mn)^2\), each component inside the parentheses is raised to the power of 2:

• 6 becomes \(6^2\)
• m becomes \(m^2\)
• n becomes \(n^2\)

This rule of powers can drastically simplify problems, making them easier to solve, especially in longer equations or more advanced problems.

Natural Numbers

Natural numbers are a fundamental concept in mathematics. They include all positive integers starting from 1, such as 1, 2, 3, and so on. These numbers are used in counting and ordering.

In algebra, natural numbers often serve as exponents in expressions. For instance, in \((6mn)^2\), 2 is a natural number, and it acts as the exponent applied to each of the terms in the expression.

Working with natural number exponents is straightforward and follows simple power rules. It's crucial in simplifying expressions since these numbers help determine how many times a number or variable will be multiplied by itself.

By understanding natural numbers, you can better predict and manage how expressions grow or reduce during simplification.

Mathematical Constants

Mathematical constants are specific numbers with fixed values used in computations, like \(\pi\) (Pi) or Euler's number \(e\). However, in basic expressions, constants are simply numbers that don’t change.

In our example, the number 6 is a constant. It’s independent of the variables and remains unchanged except for being raised to a power:

• 6 becomes \(6^2\), which simplifies to 36.

Understanding how to handle constants in expressions lets you systematically apply operations and power rules. They anchor your calculations by giving you a reference point that's stable and predictable.

This makes processing algebraic expressions less daunting by distinguishing between static numbers (constants) and changing values (variables or coefficients).

Grasping the role of mathematical constants helps in simplifying expressions accurately and efficiently.