Problem 80 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{|-25|-8(-5)}{2^{4}-29} $$

Step-by-Step Solution

Answer

The expression evaluates to -5.

Step 1: Evaluate the Absolute Value

Start by evaluating the absolute value part of the expression. The expression inside the absolute value is \(-25\) and \(|-25| = 25\).

Step 2: Multiply Inside the Expression

Next, multiply \(-8\) by \(-5\) inside the expression. This is \(-8 imes -5 = 40\).

Step 3: Simplify the Numerator

Add the results of the absolute value and multiplication to simplify the numerator. Therefore, the numerator becomes \(25 + 40 = 65\).

Step 4: Evaluate the Exponent

Calculate \(2^4\), which means \(2 imes 2 imes 2 imes 2 = 16\).

Step 5: Simplify the Denominator

Subtract 29 from the result of the exponent calculation: \(16 - 29 = -13\).

Step 6: Divide the Numerator by the Denominator

Finally, divide the numerical results calculated in the previous steps: \(\frac{65}{-13} = -5\).

Key Concepts

Absolute Value

Absolute value is a concept in mathematics that refers to the distance a number is from zero on the number line, without considering which direction from zero the number lies. In simpler terms, the absolute value of a number is its numerical value without any positive or negative sign. For example, the absolute value of -25 is written as \(|-25|\), and it equals 25.

This is because distance is always a non-negative quantity. Calculating the absolute value often comes into play when dealing with expressions containing negative numbers, as seen in the given problem. Knowing how to handle absolute values is crucial, especially when working with more complex algebraic expressions that might involve variables and unknowns.

Exponentiation

Exponentiation refers to the mathematical operation of raising a number to a power. Essentially, it means multiplying a number by itself a specified number of times. For instance, if we take \(2^4\), this means multiplying 2 by itself four times: \(2 \times 2 \times 2 \times 2 = 16\).

Understanding exponentiation is fundamental in algebra as it allows us to work with expressions involving powers efficiently.

• The base in exponential expressions is the number being multiplied, and the exponent indicates how many times to multiply the base.
• Exponential expressions frequently appear in equations and functions, making it essential for problem-solving.

It is good to practice various problems involving exponents to become comfortable with these calculations.

Multiplication in Algebra

Multiplication in algebra applies the basic principles of arithmetic multiplication but often involves variables and algebraic expressions. When you multiply numbers, you are essentially finding the total of these quantities when added together repeatedly.

In the context of the step-by-step solution given:

• The multiplication of -8 by -5 was calculated as:\(-8 \times -5 = 40\).
• Since both numbers are negative, their product is positive.

This mirrors the rule that multiplying two numbers with the same sign results in a positive product. Understanding these rules is essential, as they form the basis for manipulating and simplifying algebraic expressions more broadly.

Rational Expressions

Rational expressions are a class of algebraic expressions represented as the ratio of two polynomials. They behave similarly to fractions, as they have a numerator and a denominator which must be handled appropriately.

In this particular exercise, the expression \(\frac{|\-25|\-8(-5)}{2^4\-29}\) forms a rational expression. The steps involved in solving it include:

• Evaluating and simplifying both the numerator, \(25 + 40 = 65\), and the denominator, \(16 - 29 = -13\).
• Finally, dividing the simplified numerator by the simplified denominator to find the result, \(\frac{65}{-13} = -5\).

Grasping the concept of rational expressions helps in solving problems involving fractions and ratios in mathematics, equipping students to handle diverse mathematical tasks effectively.