Problem 8 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ 16+5^{3} $$

Step-by-Step Solution

Answer

141

Step 1: Understand the Expression

The expression to evaluate is: \[ 16 + 5^{3} \]This expression includes addition and an exponentiation. We need to calculate the exponentiation first.

Step 2: Calculate the Exponentiation

Evaluate the term with the exponent: \[ 5^{3} = 5 \times 5 \times 5 = 25 \times 5 = 125 \]Replace the exponentiation term in the original expression.

Step 3: Perform the Addition

Now that we have replaced the exponentiation term, the expression is: \[16 + 125 \]Add these two numbers together to get the final result: \[16 + 125 = 141 \]

Key Concepts

Order of Operations

When evaluating algebraic expressions like \( 16 + 5^3 \), it's essential to follow the correct sequence called the **Order of Operations**. The order is typically remembered using the acronym PEMDAS:

• **P**arentheses
• **E**xponents (including roots, like square roots)
• **M**ultiplication and **D**ivision (from left to right)
• **A**ddition and **S**ubtraction (from left to right)

This set of rules ensures that calculations are performed in the correct order, providing consistent and accurate results.
In the given exercise, following **PEMDAS** means we evaluate the exponentiation (\( 5^3 \)) before performing the addition (\( 16 + 125 \)).
So, always remember to address exponents before addition or subtraction in your expressions.
By following this order, evaluations become straightforward and error-free.

Exponentiation

**Exponentiation** is a fundamental operation in algebra. It’s the process of raising a number to the power of another number.
For instance, in \( 5^3 \), the base is 5, and the exponent is 3. This tells us to multiply 5 by itself three times: \( 5 \times 5 \times 5 \).
Following these steps:

• First, calculate \( 5 \times 5 = 25 \)
• Then, multiply the result by 5 again: \( 25 \times 5 = 125 \)

So, \( 5^3 = 125 \).
Replacing \( 5^3 \) in the original expression \( 16 + 5^3 \) gives us \( 16 + 125 \), simplifying further to our final result.

Addition in Algebra

Once exponentiation is complete, we move on to **Addition**, which is the final operation here.
In algebra, addition works the same as with regular numbers. You simply add the values together.
For the expression \( 16 + 125 \):

• Start with 16
• Add 125 to it: \( 16 + 125 = 141 \)

Thus, \( 16 + 125 \) equals 141.
Always ensure to re-check your additions to avoid minor errors.
Mastering simple addition in algebra sets the foundation for solving more complex equations involving multiple terms and operations.