Problem 52 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ 8^{2}+3+5(2+7) $$

Step-by-Step Solution

Answer

**Question:** Simplify the expression: $$ 8^{2} + 3 + 5(2 + 7). $$ **Answer:** 112

Step 1: Identify the elements in the expression according to the order of operations

In the given expression, $$ 8^{2} + 3 + 5(2 + 7), $$ we can identify the following items: - Exponent: \(8^2\) - Parentheses: \((2+7)\) - Multiplication: \(5 \times (2+7)\) - Addition: \(8^2 + 3 + 5 \times (2+7)\).

Step 2: Evaluate the exponent

According to the order of operations, we first evaluate the exponent: $$ 8^2 = 64. $$ So now the expression becomes: $$ 64 + 3 + 5(2 + 7). $$

Step 3: Evaluate the expression inside the parentheses

According to the order of operations, we evaluate the expression inside the parentheses: $$ (2 + 7) = 9. $$ So now the expression becomes: $$ 64 + 3 + 5(9). $$

Step 4: Perform the multiplication

According to the order of operations, we perform the multiplication: $$ 5(9) = 45. $$ So now the expression becomes: $$ 64 + 3 + 45. $$

Step 5: Perform the addition

According to the order of operations, we perform the addition: $$ 64 + 3 + 45 = 112. $$ Thus, the simplified expression is equal to \(\boldsymbol{112}\).

Key Concepts

Simplify Mathematical Expressions

When we talk about simplifying mathematical expressions, we refer to the process of reducing a complex equation or formula to its simplest form. This means combining like terms, carrying out arithmetic operations, and removing any unnecessary parentheses where possible. Simplifying an expression makes it easier to understand and work with.

For example, consider the expression 8² + 3 + 5(2 + 7). The goal is to simplify this expression step by step until we reach the most reduced form.

• The first step is to identify and perform any operations within the parentheses as these have the highest priority.
• Next, we address any exponents in the expression.
• Following that, we carry out the multiplication and division operations from left to right.
• Finally, we perform addition and subtraction, also from left to right.

This systematic approach ensures that all mathematicians will simplify expressions consistently, leading to the same final result.

Exponents

Exponents represent repeated multiplication and are a fundamental concept in algebra and more advanced mathematics. An exponent is written as a small number to the right and above a base number and tells how many times to multiply the base by itself.

For instance, in the expression 8², the number 8 is the base and the number 2 is the exponent, indicating that 8 should be multiplied by itself once: 8 × 8 = 64.

It is essential to evaluate the exponents early in the simplification process, directly after any calculations needed for quantities within parentheses. This is because exponents can significantly alter the value of numerical expressions and have a higher priority than multiplication, division, addition, or subtraction.

Parentheses in Algebra

Parentheses play a crucial role in algebra. They indicate which operations should be performed first in an expression. Without clear rules regarding the use of parentheses, expressions could be interpreted in multiple ways, leading to different and often incorrect results.

In our example, 5(2 + 7), the parentheses tell us to add 2 and 7 together before doing anything else. Once we calculate the sum within the parentheses, we then treat it as a single number in the expression. This intermediate step prevents confusion and ensures the integrity of the mathematical operations as prescribed by the order of operations.

Understanding how to correctly work with parentheses can prevent common errors and is instrumental when dealing with more complex expressions, where multiple sets of parentheses might be involved.