Problem 63 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ \frac{(1+6)^{2}+2}{19} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $\frac{(1+6)^{2}+2}{19}$ Answer: $2\frac{13}{19}$

Step 1: Parentheses/Brackets

Evaluate the expression inside the parentheses: $$ (1 + 6) = 7 $$

Step 2: Exponents/Orders

Raise the result from Step 1 to the power of 2: $$ 7^{2} = 49 $$

Step 3: Addition

Add the result from Step 2 to 2: $$ 49 + 2 = 51 $$

Step 4: Division

Divide the result from Step 3 by 19: $$ \frac{51}{19} = 2\frac{13}{19} $$ The simplified expression is: $$ 2\frac{13}{19} $$

Key Concepts

Parentheses

Understanding the role of parentheses in mathematical expressions is crucial. Parentheses indicate the order in which operations should be performed. In our exercise, the parentheses enclose the numbers 1 and 6. According to the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), the calculation inside the parentheses must be completed first.

So, in the expression \((1+6)^{2}+2\), we begin by adding the numbers inside the parentheses: \(1 + 6 = 7\). Only after this step do we proceed to the next stages of the operation. Remember, neglecting the proper order can lead to incorrect results, so pay careful attention to parentheses when simplifying expressions.

Exponents

After dealing with parentheses, we tackle the exponents. Exponents are a shorthand way to represent repeated multiplication of the same number. For example, the expression \(7^{2}\) from our exercise means \(7 \times 7\), which equals 49.

Exponents take priority over basic operations like addition, subtraction, and simple multiplication or division not involving parentheses. It's essential to accurately calculate the value of exponential expressions before moving on to these other operations to simplify the expression correctly.

Simplify Expressions

Combining Operations

After calculating the values inside parentheses and exponents, we proceed to simplify expressions by performing addition, subtraction, multiplication, and division in order. In our example, we are left with \(49 + 2\) after resolving the parentheses and exponents. Because there are no multiplication or division operations to perform before addition, we simply add the two numbers together to get 51.

During this step, we must be careful not to hurry through calculations. Each operation is like a building block; missing one could make the entire structure—or in our case, the final answer—unstable.

Algebraic Fractions

When the expression to be simplified involves an algebraic fraction—that is, a fraction that includes variables or expressions—we still follow the order of operations. However, the numerator and the denominator of the fraction may have their own sets of operations to be evaluated separately first. Since our example has a fraction \(\frac{51}{19}\), we simply divide the numerator by the denominator. In the case where the divisor doesn't perfectly divide the dividend, we could be left with a remainder, as represented in mixed number format. Thus, 51 divided by 19 gives us a whole number result of \(2\), which is the final simplified form of the expression we were given to simplify.