Problem 21 - College Algebra with Corequisite Support

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: Evaluate the Power

First, evaluate the exponent in the expression. The expression is \(2^2\), which equals \(4\). Thus, the expression becomes \(9 + 4(4)\).

Step 2: Multiply

Next, multiply \(4\) by the result of \(2^2\), which we calculated as \(4\). Therefore, perform the multiplication \(4 \times 4 = 16\). The expression now simplifies to \(9 + 16\).

Step 3: Add

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

Key Concepts

Order of Operations

In mathematics, there's a specific set of rules to follow called the order of operations. This ensures calculations are done in a standardized way to get the correct result. A common way to remember this order is with the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

These operations must be followed in this sequence to correctly simplify expressions. Parentheses and exponents take priority over other operations. Multiplication and division come next, and are on the same level of priority; these should be performed from left to right in the expression. Finally, perform addition and subtraction, also from left to right. In the case of the original exercise, this order was followed to simplify the expression correctly.

Exponents

Exponents are used to express repeated multiplication of the same number. The exponent, typically written as a superscript number, tells you how many times the base number is multiplied by itself. For instance, in the expression \(2^2\), the base is \(2\) and the exponent is \(2\), which means \(2\) multiplied by itself: \(2 \times 2\), resulting in \(4\).
When simplifying expressions with exponents, it is crucial to evaluate the exponents first (after resolving any parentheses) according to the order of operations.
In the given exercise, \(2^2\) was calculated first to simplify the expression accurately, leading to further steps.

Arithmetic Operations

Arithmetic operations are the basic tasks of mathematics that include addition, subtraction, multiplication, and division. Once the higher-priority operations like exponents are handled, as per the order of operations, the arithmetic operations are performed to simplify the expression completely.
In the expression \(9+4\left(2^{2}\right)\), after solving the exponent \(2^2\), multiplication is performed next: \(4 \times 4\) yielding \(16\).
This multiplication is part of simplifying the expression by breaking it down into simpler arithmetic tasks. Finally, the resulting number, \(16\), is added to \(9\), completing the simplification: \(9 + 16 = 25\). Understanding these basic operations helps in handling more complex mathematical problems efficiently.