Problem 57 - Elementary and Intermediate Algebra

Question

Simplify. Match the algebraic expression with the equivalent rewritten expression below. Check your answer by calculating the expression by hand and by using a calculator. A) \((5(3-7)+4 \wedge 3) /(-2-3)^{2}\) B) \((5(3-7)+4 \wedge 3) /\left(-2-3^{2}\right)\) C)\((5(3-7)+4) \wedge 3 /-2-3^{2}\) D) \(5(3-7)+4 \wedge 3 /(-2-3)^{2}\) $$ \frac{5(3-7)+4^{3}}{(-2-3)^{2}} $$

Step-by-Step Solution

Answer

Option A

Step 1 - Simplify inside parentheses

First, simplify the expression inside the parentheses. For the given expression, compute: \(3-7 = -4\) and \(-2-3 = -5\).

Step 2 - Simplify exponents

Next, calculate the exponents: \(4^3 = 64\) and \((-5)^2 = 25\).

Step 3 - Simplify multiplication and addition

Then, evaluate the multiplication and addition: \(5(-4) + 64 = -20 + 64 = 44\).

Step 4 - Compute the division

Finally, divide the simplified expression in the numerator by the simplified expression in the denominator: \( \frac{44}{25} \).

Step 5 - Match with given choices

Compare the calculated expression \( \frac{5(3-7) + 4^3}{(-2-3)^2} = \frac{44}{25} \) with the given options. Option A) \( \frac{5(3-7) + 4^3}{(-2-3)^2} \) matches our calculation.

Key Concepts

Order of Operations

To solve algebraic expressions correctly, you must follow a specific order of operations. This sequence is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Following this order ensures that everyone solves the equation consistently and correctly.

When simplifying the given expression, start with the innermost parentheses, then move to exponents, and then handle any multiplication or division. Finally, complete any addition or subtraction. Sticking to this hierarchy avoids mistakes and ensures accuracy.

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. For example, in the expression \(4^3\), the number 4 is multiplied by itself three times (i.e., \(4 \times 4 \times 4\)).

For the given exercise, we encounter exponents like \(4^3\) and \((-5)^2\). Solving these correctly involves precise calculation:

• \(4^3 = 64\)
• \((-5)^2 = 25\)

Parentheses Simplification

Simplifying expressions inside parentheses is crucial because it alters the values you work with in subsequent steps. Parentheses dictate the priority of operations, making sure some calculations are done first.

In our provided problem, the first step involves simplifying within the parentheses. We compute:

• \(3-7 = -4\)
• \(-2-3 = -5\)

By handling these parts first, we obtain simpler values to work with in the later steps.

Fraction Simplification

Fractions represent one quantity divided by another. Simplifying the numerator and denominator separately makes the expression easier to handle.

In the given exercise, after simplifying within parentheses and calculating exponents, the fraction becomes \(\frac{5(-4) + 64}{25}\). Simplify this fraction by evaluating the numerator and denominator:

• Numerator: \(5(-4) + 64 = -20 + 64 = 44\)
• Denominator: \((-5)^2 = 25\)

Finally, we get a simplified fraction: \(\frac{44}{25}\).

Algebraic Expressions

Algebraic expressions combine numbers and variables with arithmetic operations. Variables represent unknown values and can assume different numbers. Simplifying these expressions involves using the order of operations to evaluate the expression step by step.

For the provided problem, you simplify the algebraic expression: \(\frac{5(3-7) + 4^3}{(-2-3)^2}\). Follow through each step we discussed:

• Handle parentheses: \(3-7 = -4\) and \(-2-3 = -5\)
• Solve exponents: \(4^3 = 64\) and \((-5)^2 = 25\)
• Simplify the numerator: \(5(-4) + 64 = -20 + 64 = 44\)
• Simplify the fraction: \(\frac{44}{25}\)

By working through each component in order, we achieve the final simplified expression.