Problem 37 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. \((8-3)^{2}+\left(2+3^{2}\right)^{2}\)

Step-by-Step Solution

Answer

The value of the expression is 146.

Step 1: Solve Inside Parentheses

First, address the expressions inside the parentheses: 1. \[8 - 3 = 5\]2. \[3^2 = 9\] and then \[2 + 9 = 11\]. Thus, simplify the expression to \((5)^2 + (11)^2\).

Step 2: Exponents

Now, solve the exponents for each term:1. \[5^2 = 25\]2. \[11^2 = 121\]. So the expression simplifies to \[25 + 121\].

Step 3: Addition

Finally, add the results of the previous step:\[25 + 121 = 146\].

Key Concepts

parentheses

In mathematics, parentheses play a crucial role in organizing and prioritizing expressions. They indicate that the operations within them should be performed first, adhering to the order of operations rules. It's similar to a set of instructions that tell you what to handle initially before moving to other tasks.

When you encounter an equation like \[ (8-3)^{2}+(2+3^{2})^{2} \], the parentheses command you to first solve whatever is inside them. This ensures clarity and correctness, preventing any mix-up with calculations outside the parentheses. Treat each parenthesis as a puzzle that needs solving before tackling the rest.

• In the first part, \(8 - 3 = 5\).
• In the second, compute \(3^2 = 9\) before adding it to 2, which gives \(2 + 9 = 11\).

Once everything inside the parentheses is simplified, you can move on to the next steps in the expression.

exponents

After simplifying any operations within parentheses, the next step is to deal with exponents. Exponents are a way to express repeated multiplication of the same number. This operation is considered before multiplication or division when solving equations.

Consider the expression \(5^2\). This tells you to multiply 5 by itself, resulting in 25. Similarly, for \(11^2\), you multiply 11 by itself to get 121. Exponents significantly change the value of numbers and are essential for more complex calculations involving growth or area, for example.

• Remember: an exponent indicates how many times to use the number in a multiplication.
• Always respect the order to avoid errors in complex calculations.

addition

Once all the operations involving parentheses and exponents have been tackled, addition comes into play. Addition is one of the basic arithmetic operations and involves combining values to get a total sum.

In the example, you add the results of your exponents: \(25 + 121\). Each of these numbers results from the computations involving parentheses and exponents beforehand. By adding them together, you arrive at the final result, 146.

• Addition is straightforward once previous steps are accurately completed.
• It's essential to follow each step in the order of operations to ensure a correct final answer.