Problem 91 - Prealgebra

Question

Use the rule for order of operations to simplify each expression. $$3 \cdot 5^{2}-75 \div 5+2^{3}$$

Step-by-Step Solution

Answer

The simplified expression is 68.

Step 1: Handle Exponents

First, calculate the expressions with exponents in the expression. We have two exponents: \(5^2\) and \(2^3\). Calculate \(5^2 = 25\) and \(2^3 = 8\). Rewrite the expression with these values: \(3 \cdot 25 - 75 \div 5 + 8\).

Step 2: Perform Multiplication and Division

Now, perform multiplication and division from left to right. Start with the multiplication: \(3 \cdot 25 = 75\). The expression becomes \(75 - 75 \div 5 + 8\). Next, do the division: \(75 \div 5 = 15\). Now, the expression is \(75 - 15 + 8\).

Step 3: Perform Addition and Subtraction

Perform addition and subtraction from left to right. First, calculate \(75 - 15 = 60\). Then, add \(60 + 8 = 68\).

Key Concepts

Exponents

Exponents are used to represent repeated multiplication of the same number. For example, in the expression \( 5^2 \), the base number \( 5 \) is multiplied by itself, resulting in \( 5 \times 5 \). This equals \( 25 \). Similarly, for \( 2^3 \), the base number \( 2 \) is used three times in multiplication: \( 2 \times 2 \times 2 = 8 \).

When simplifying expressions with exponents, it's important to handle all exponentiation first before moving on to other operations. This is because exponents have a higher priority in the order of operations.

• Identify all parts of an expression with exponents.
• Complete these calculations first.
• Replace the original exponent part with the calculated value.

Multiplication and Division

After solving any exponents, focus on multiplication and division. These two operations are considered as equal in the hierarchy and should be processed from left to right in the expression.

For example, in the expression \( 3 \cdot 25 - 75 \div 5 + 8 \):

• First, perform the multiplication \( 3 \times 25 \), which equals \( 75 \).
• Next, handle the division \( 75 \div 5 \), which results in \( 15 \).
• Substitute these calculated values back into the expression.

The key is to move sequentially from the beginning to the end of the expression, handling these operations as they appear.

Addition and Subtraction

Finally, after completing the exponents, multiplication, and division, you will handle addition and subtraction. These operations also have equal status in terms of priority, so they should also be processed from left to right, just like multiplication and division.In our example, the expression at this point is \( 75 - 15 + 8 \).

• Start with the first operation: \( 75 - 15 = 60 \).
• Then proceed with the final addition: \( 60 + 8 = 68 \).
• This results in the simplified expression value of \( 68 \).

By following the steps, the correct solution will be obtained efficiently and accurately. Remember, moving left to right is crucial when handling addition and subtraction to prevent any mistakes.