Problem 58 - Fundamentals of Mathematics

Question

Use the order of operations to simplify \(\left(5^{2}+3^{2}+2\right) \div 2^{2}\).

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Exponentiation

First, solve any exponents in the expression. We have two exponent terms: \(5^2\) and \(3^2\).\[5^2 = 25\3^2 = 9\]

Step 2: Parentheses Simplification

Now substitute the results of the exponents back into the expression and simplify inside the parentheses.\[(25 + 9 + 2)\]Add the numbers together inside the parentheses:\[25 + 9 + 2 = 36\]

Step 3: Division

Next, handle the division by evaluating \(2^2\):\[2^2 = 4\]Now perform the division using the result from the parentheses:\[36 \div 4 = 9\]

Key Concepts

Exponentiation

Exponentiation is a fundamental concept in mathematics that involves raising a number, known as the base, to the power of an exponent. When you see a number with a smaller number up above it, like in the expressions \(5^2\) or \(3^2\), this means you have to multiply the base number by itself as many times as the exponent indicates. For example:

• \(5^2\) means 5 multiplied by itself 2 times: \(5 \times 5 = 25\).
• Similarly, \(3^2\) means 3 multiplied by itself: \(3 \times 3 = 9\).

Using exponentiation as the first step means you simplify these operations to make the whole expression easier to handle. It also ensures that calculations follow a consistent and predictable pattern, preventing any potential errors from attempting to process operations out of order. By tackling exponents first, we lay the groundwork to simplify the entire expression methodically.

Parentheses Simplification

The principle of simplifying inside parentheses is critical for resolving expressions step-by-step. Parentheses indicate that operations inside should be performed before anything else outside them. This ensures clarity in complex calculations and avoids mistakes. In the expression \((5^2 + 3^2 + 2)\), once we've handled exponentiation, it becomes \((25 + 9 + 2)\).Now, it's time to add the numbers together inside the parentheses:

• First, add 25 and 9, which results in 34.
• Then add 2 to 34, resulting in 36.

Simplifying inside the parentheses gives \(36\). This step highlights how grouping numbers and operations helps us manage and simplify sections of an expression systematically. Once everything inside the parentheses is simplified, we can proceed to the next operations outside of them.

Division

Division is often one of the final steps in solving an expression like this, especially when using the order of operations. Once we've simplified inside the parentheses to get 36, our next task is to divide by the result of any remaining or simplified operations outside it.In this example, after calculating the exponents and simplifying inside the parentheses, we are left with the division part, \(36 \div 2^2\). Remember, before dividing, ensure that you handle any exponents in the divisor. Here, \(2^2\) is 4, so the expression becomes:

• \(36 \div 4 = 9\).

Executing division carefully ensures the accuracy of your solutions. Solving expressions systematically, by following the order of operations, guarantees you reach the correct answer efficiently, reducing the possibility of errors usually encountered in manual calculations.