Problem 52 - College Algebra

Question

For the following exercises, simplify the expression. $$ 5^{2}-4(3 x) $$

Step-by-Step Solution

Answer

The simplified expression is \(-12x + 25\).

Step 1: Simplify the power

First, simplify the expression by calculating the power. \(5^2\) means multiplying 5 by itself, so \(5^2 = 25\). Now, the expression becomes \(25 - 4(3x)\).

Step 2: Distribute the multiplication

Next, distribute the 4 across the term in the parenthesis. Multiply \(4\) by \(3x\) to get \(12x\). So, the expression now is \(25 - 12x\).

Step 3: Arrange the expression

Finally, write the expression in standard form, which is usually the term with the variable first, followed by constants. Therefore, the expression \(25 - 12x\) becomes \(-12x + 25\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is all about making a mathematical expression easier to understand and work with. The goal is to reduce the expression into its most straightforward form without changing its value. This is done by applying basic arithmetic operations and following algebraic rules.

There are a few key things to remember when simplifying expressions:

• Look for options to combine like terms. A lot of times expressions have similar terms that can be added or subtracted from one another.
• Carry out any indicated operations such as powers, multiplication, or division within the expression first, following the order of operations.
• Always check if the expression can be further simplified or needs rearranging for easier interpretation.

In the exercise provided, we are simplifying the expression by addressing the power first, distributing multiplication, and then presenting it in a standard format.

Distribution in Algebra

Distribution in algebra involves expanding expressions to simplify them. This is done using the distributive property which is all about distributing or spreading out multiplication over addition or subtraction inside parentheses. It follows the rule: \( a(b + c) = ab + ac \).

In our problem, the expression is \(25 - 4(3x)\), and the term \(4(3x)\) requires distribution. Here's how it works:

• Take the "outside" number, in this case, 4, and multiply it by each term inside the parentheses.
• For this expression, multiply 4 by \(3x\) to get \(12x\).
• After doing this, the expression becomes \(25 - 12x\).

This helps in getting rid of the parentheses, making the expression simpler and more manageable for further simplification or solving.

Powers in Algebra

In algebra, powers, also called exponents, simplify expressions by denoting repeated multiplication of a number by itself. For instance, \( a^n \) means multiplying \(a\) by itself \(n\) times.

This is often one of the first steps when simplifying expressions. For the given exercise, the power involved is \(5^2\). Here's the process for simplifying it:

• Recognize that \(5^2\) means \(5\times5\).
• Calculate the product, which results in 25.
• Replace the original expression \(5^2\) in the problem with 25, simplifying one part of the expression right away.

Dealing with powers early on can simplify other operations that need to happen. By resolving powers first, the rest of the algebraic simplification becomes more straightforward.