Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$(3 \cdot 5)^{2}-3 \cdot 5^{2}$$

Step-by-Step Solution

Answer

The simplified result of \((3 \cdot 5)^{2}-3 \cdot 5^{2}\) is 150.

Step 1: Parentheses

First, perform the operations inside parentheses. So, for both \((3 \cdot 5)^{2}\) and \(3 \cdot 5^{2}\), calculate \(3 \cdot 5\). This gives *\((3 \cdot 5)^{2} = 15^{2}\)* and *\(3 \cdot 5^{2} = 3 \cdot 25\)*.

Step 2: Exponents

Then, calculate the exponents. \(15^{2} = 225\) and \(3 \cdot 25 = 75\).

Step 3: Subtraction

Finally, subtract the values obtained for \(3 \cdot 5^{2}\) from \((3 \cdot 5)^{2}\). So, \(225 - 75 = 150\).

Key Concepts

Exponents

Exponents can be seen as repeated multiplication. This mathematical operation tells us how many times to multiply a number by itself. For example, in the expression \(15^2\), the number 15 is the base, and the 2 is the exponent. This means we multiply 15 by itself, giving us:

• \(15 imes 15 = 225\)

Exponents are very important in calculations, as they significantly impact the value of an expression. By following the order of operations, exponents should always be calculated after parentheses but before multiplication, division, addition, or subtraction. Having a grasp on how exponents work is essential for simplifying more complex mathematical expressions effectively.

Parentheses

Parentheses are a critical component in mathematics, often used to indicate which operations should be performed first. According to the order of operations, any calculation inside parentheses takes priority over others. It provides clarity and ensures that calculations are performed correctly. For instance, in the expression \((3 \cdot 5)^{2}\), the operation \(3 \cdot 5\) needs to be calculated first inside the parentheses:

• \(3 \cdot 5 = 15\)

This result is then used as the base of the exponent. By performing operations inside parentheses first, we preserve the intended grouping of numbers and operations, ensuring accurate calculations. Parentheses help control the flow of calculations and are essential in following the correct order of operations.

Simplifying Expressions

Simplifying expressions involves breaking down a complex expression into its simplest form. This means performing all possible calculations, reducing the expression to a single number or a smaller set of operations. The order of operations, often remembered by the acronym PEMDAS, guides the process of simplification:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our example, we started by solving what's inside the parentheses, followed by dealing with exponents, and then performing the final subtraction. This systematic approach helps ensure no errors occur in a calculation and maintains the integrity of the mathematical expression being addressed.