Problem 6 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ 100-7^{2} $$

Step-by-Step Solution

Answer

51

Step 1 - Understand the Problem

The exercise requires processing a mathematical expression: $$100 - 7^{2}$$ First, note that the expression involves subtracting the square of 7 from 100.

Step 2 - Calculate the Square of 7

Evaluate the square of 7. Use the formula for squaring a number: $$7^2 = 7 \times 7$$ This gives us: $$7^2 = 49$$

Step 3 - Subtract the Result from 100

Subtract 49 from 100. Perform the subtraction: $$100 - 49 = 51$$

Key Concepts

Evaluating Expressions

In elementary algebra, evaluating expressions involves finding the value of an algebraic expression by substituting numbers in place of variables and performing arithmetic operations. In the exercise we have: \(100 - 7^{2}\), there are no variables, so we only need to complete the operations in the given order.

• Order of Operations: When evaluating any expression, always follow the order of operations, often remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

For this specific expression, the crucial operation is the exponentiation. After dealing with the exponent, the final step will be a simple subtraction. Let's explore these operations in more detail.

Exponents

Exponents represent repeated multiplication of the same number. For instance, \(7^2\) means 7 multiplied by itself. This is expressed as: \[7 \times 7\ = 49\].

• Understanding Exponents: The exponent (2 in this case) tells you how many times to multiply the base number (7) by itself.

For example, \(2^3 = 2 \times 2 \times 2 = 8\). Knowing how to handle exponents is essential in evaluating expressions because it is one of the first operations you perform according to PEMDAS. After calculating \(7^2 = 49 \) in our exercise, we then move on to the next step which involves subtraction.

Subtraction

Subtraction is one of the basic arithmetic operations where we remove a number from another number. After solving the exponent part of the expression in the exercise, we are left with a simple subtraction: \(100 - 49\).

• Carrying Out Subtraction: Start by ensuring that the numbers are aligned correctly if working on paper. Subtract each digit starting from the rightmost digit (units place) to the left.

In this problem, we subtract 49 from 100:
\[ 100 - 49 = 51 \]. Therefore, the result of the given expression \(100 - 7^{2} \) is 51.
This thorough understanding of each step helps in mastering elementary algebra and confidently evaluating more complex expressions.