Problem 74 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{13^{2}-5^{2}}{-3\left(5-3^{2}\right)} $$

Step-by-Step Solution

Answer

The value of the expression is 12.

Step 1: Simplify the Inner Expression

First, simplify the expression inside the parentheses: \[5 - 3^2\]Calculate \(3^2\), which equals 9.\[5 - 9 = -4\]So, the expression inside the parentheses simplifies to \(-4\).

Step 2: Evaluate the Numerator

Next, we need to evaluate the numerator:\[13^2 - 5^2\]Calculate \(13^2\), which is 169, and \(5^2\), which is 25.Subtract these values:\[169 - 25 = 144\]

Step 3: Evaluate the Denominator

Now evaluate the denominator:\[-3(-4)\]Multiply \(-3\) and \(-4\) to get:\[12\]

Step 4: Final Division

Finally, divide the simplified numerator by the simplified denominator:\[\frac{144}{12} = 12\]

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of another number, called the exponent. In simpler terms, it means multiplying the base by itself as many times as indicated by the exponent. For example, in the expression \(3^2\), the base is 3 and the exponent is 2, which means you multiply 3 by itself once, resulting in 9.

In our exercise, we have two numbers raised to powers: \(13^2\) and \(5^2\). Let's break them down:

• \(13^2\): Multiply 13 by itself, resulting in 169.
• \(5^2\): Multiply 5 by itself, resulting in 25.

Expressing numbers using exponentiation not only makes calculations clearer but is also more efficient, especially with large numbers. Understanding how to work with exponents is crucial when simplifying or solving algebraic expressions.

Order of Operations

The Order of Operations is an essential rule in mathematics that dictates the sequence in which operations should be performed to accurately solve an expression. An easy way to remember the sequence is the acronym PEMDAS:

• Parentheses first
• Exponents next
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our exercise, let's apply these steps:

• First, solve expressions within the parentheses: \(5 - 3^2\). Calculate the exponent first (\(3^2 = 9\)), then perform the subtraction: \(5 - 9 = -4\).
• Next, handle any remaining exponents, as seen in \(13^2 - 5^2\).
• Then, conduct multiplication, as with \(-3(-4)\).
• Finally, perform any division: \(\frac{144}{12} = 12\).

By adhering to the order of operations, we ensure each calculation is precise, leading us to the correct final result.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic and concise form, often making them easier to work with and understand. This process combines several key mathematical operations, adhering to the order of operations to maintain accuracy.

In the given exercise, simplification occurs in several steps:

• First, simplify within the parentheses: \(5 - 3^2\) results in \(5 - 9 = -4\).
• Next, simplify the other parts of the expression, such as the numerator: \(13^2 - 5^2\) gives \(169 - 25 = 144\).
• Similarly, simplify the denominator through multiplication: \(-3(-4) = 12\).
• Finally, divide the simplified numerator by the simplified denominator to complete the expression: \(\frac{144}{12} = 12\).

By progressively reducing the expression with each operation, simplifying provides a clearer path to the solution, revealing the simplest form of the original expression. This method not only ensures accuracy but also enhances a deeper understanding of algebraic manipulation.