Problem 78 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ \frac{(10-4)^{2}}{14-5} $$

Step-by-Step Solution

Answer

4

Step 1: Subtract inside the parentheses

Calculate the value inside the parentheses: \(10 - 4 = 6\)

Step 2: Square the result from Step 1

Square the result obtained in Step 1: \(6^{2} = 36\)

Step 3: Subtract the denominator

Calculate the value of the denominator: \(14 - 5 = 9\)

Step 4: Divide the numerator by the denominator

Divide the squared value by the denominator: \(\frac{36}{9} = 4\)

Key Concepts

Order of Operations

Math can sometimes be confusing, especially when you have multiple operations in a single expression. To solve such problems correctly, you have to follow the 'Order of Operations.' The Order of Operations is a set of rules that dictates the sequence in which you should perform different mathematical operations.

Remember the acronym PEMDAS to help you remember the sequence: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following these rules will ensure you get the correct result every time.

Parentheses

Parentheses are like markers telling you to deal with what’s inside them first. In our example \( \frac{ (10-4)^{2} }{ 14-5 } \), we look at what’s inside the parentheses first: \(10-4\).

Calculate: \(10 - 4 = 6\). Only after solving everything inside the parentheses can you move on to other operations. This ensures that you maintain accuracy in your calculation. Always remember - deal with parentheses first!

Exponents

Exponents come next in our Order of Operations. They tell you how many times to multiply a number by itself. In our example, after solving the parentheses (\(10-4 = 6)\), you then square the result: \(6^2\).

Calculate: \( 6 \times 6 = 36 \). Exponents are important for showing repeated multiplication. Always make sure you handle them after you’ve dealt with parentheses but before doing any multiplication, division, addition, or subtraction.

Division

Finally, we come to division. According to our Order of Operations, after dealing with parentheses and exponents, we handle multiplication and division from left to right. In our case, after calculating \(36\) in the numerator and \(14-5 = 9\) in the denominator, we divide:

Calculate: \(\frac{36}{9} = 4\). This gives us the final answer. Always remember to follow the Order of Operations to get correct results in mathematical expressions.