Problem 69 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{2 \cdot 2^{5}-60+(-4)}{5^{4}-(-4)(-5)} $$

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Simplify the numerator

The numerator of the expression is given by \( 2 \cdot 2^{5} - 60 + (-4) \). First, calculate \( 2^{5} \): \[2^{5} = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32\]Then, multiply by 2:\[2 \cdot 32 = 64\]Now, substitute back into the numerator expression:\[64 - 60 + (-4) = 64 - 60 - 4 = 0\]So, the numerator simplifies to 0.

Step 2: Simplify the denominator

The denominator of the expression is given by \( 5^{4} - (-4)(-5) \). First, calculate \( 5^{4} \):\[5^{4} = 5 \cdot 5 \cdot 5 \cdot 5 = 625\]Now compute \((-4) \times (-5)\):\[(-4) \times (-5) = 20\]Finally, substitute back into the denominator expression:\[625 - 20 = 605\]The denominator simplifies to 605.

Step 3: Divide the simplified numerator by the simplified denominator

Now that you have simplified both parts, substitute the simplified values back into the expression:\[\frac{0}{605} = 0\] Any number divided by 605 is zero, provided the numerator is zero.

Key Concepts

Numerical Operations

Numerical operations are foundational in evaluating algebraic expressions. They encompass all basic arithmetic operations such as addition, subtraction, multiplication, and division. When evaluating an expression like \(2 \cdot 2^{5} - 60 + (-4)\), each operation plays a crucial role. For instance:

• Multiplication: In the expression, multiplying 2 by \(2^5\) is a key numerical operation.
• Addition/Subtraction: Adding and subtracting values like \(-60\) and \(-4\) alter the expression's total.

These operations form the backbone of further arithmetic and algebraic processes. They are instrumental in systematically breaking down complex expressions into simpler parts.
Managing these effectively requires a clear understanding of order and precedence to avoid errors, making them indispensable in problem-solving scenarios.

Exponents

Exponents are a powerful tool in mathematics that describes how many times a number, known as the base, is multiplied by itself. In the expression provided, understanding exponents is essential. The term \(2^5\) signifies that 2 is multiplied by itself 5 times:

• Understanding Exponents: This results in the calculation \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32\).
• Application: Exponents simplify repeated multiplication, allowing for compact representation and easier calculation.

Using exponents efficiently reduces the complexity of problems, especially as numbers grow larger. Moving on to the denominator, we see another exponent \(5^4\), which involves multiplying 5 by itself 4 times, resulting in 625. Recognizing and manipulating exponents accurately is crucial in both algebra and more advanced fields of math.

Step-by-Step Simplification

Step-by-step simplification is a systematic process to break down complex expressions into manageable pieces. With our expression \(\frac{2 \cdot 2^{5}-60+(-4)}{5^{4}-(-4)(-5)}\), an orderly approach helps ensure precision and clarity. Here's how the process unfolds:

• Simplifying the Numerator: First, calculate \(2 \cdot 2^5\) to obtain 64, then perform the operations \(64 - 60 - 4\) to get a simplified result of 0.
• Simplifying the Denominator: Compute \(5^4\) which equals 625, then subtract \((-4)(-5) = 20\) to achieve 605.
• Final Division: With a simplified numerator and denominator, solve \(\frac{0}{605}\), which equals 0.

Through step-by-step simplification, each part of an expression gets the attention it needs, minimizing errors and illustrating logical clarity in problem-solving. This methodical path from start to finish is invaluable in both learning and applying mathematical concepts.