Problem 21 - College Algebra

Question

For the following exercises, simplify the given expression. $$ 9+4\left(2^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is 25.

Step 1: Solve the Exponent

First, solve the exponent inside the parentheses. We have \(2^2\), which means we multiply 2 by itself: \(2 \times 2 = 4\). The expression inside the parentheses now becomes 4.

Step 2: Multiply by the Coefficient

Next, multiply the result from the exponent by the coefficient in front of the parentheses. So, we multiply 4 by 4 (the coefficient): \(4 \times 4 = 16\).

Step 3: Add Remaining Terms

Finally, add 9 (from the beginning of the expression) to the result of the multiplication: \(9 + 16 = 25\). The simplified value of the expression is 25.

Key Concepts

Order of Operations

When simplifying any mathematical expression, following the correct order of operations is crucial. It's like having a recipe to follow, ensuring that every step leads to the desired outcome. The universally accepted order is remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots)
• MD: Multiplication and Division (from left to right)
• AS: Addition and Subtraction (from left to right)

You always start with the innermost parentheses and work your way outward, handling exponents next. This ensures that calculations are performed correctly and consistently. For the expression, we began with solving the exponent inside the parentheses because that's the second rule after evaluating parentheses.

Exponents

Exponents are an efficient way of expressing repeated multiplication. In our exercise, we encountered the term \(2^2\). This denotes that 2 is used as a factor twice, or \(2 \times 2\).
Here's why understanding exponents is vital:

• They simplify complex multiplication.
• They appear frequently in algebra, especially in polynomial expressions and equations.

Our step involved computing \(2^2\), which equals 4, thus simplifying the expression inside the parentheses. By mastering exponents, you can tackle various mathematical problems more effectively.

Coefficients

Coefficients play a critical role in algebra. They are the numerical factor in front of a term, often found in expressions involving variables or constants. In our exercise, the term \(4(2^2)\) had a coefficient of 4.
Let’s break down why coefficients are important:

• They signify how many times a term is multiplied.
• They allow for easy manipulation and simplification of expressions.

This particular step involved multiplying the coefficient 4 by the result of the exponentiation, converting 4 times 4 to equal 16. Understanding coefficients helps in quickly evaluating and simplifying expressions in algebra.

Addition in Algebra

Addition is one of the simplest yet fundamental operations in algebra. It's the process of combining two or more numbers. In the final step of our expression, after simplifying the other operations, we added 9 to 16.
Here's what makes addition vital:

• It helps combine terms to reach a final result.
• It's essential in forming equations and solving for unknowns.

By understanding addition, you are equipped to combine simplified expressions effectively, leading to the final solution of any algebraic problem. In our expression, adding was the last step, bringing us to the answer 25.