Problem 41 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(\frac{6+|8-2|+3^{2}}{18-3}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{7}{5} \).

Step 1: Evaluate the Absolute Value

Evaluate the expression inside the absolute value, \[|8-2| = |6| = 6.\]

Step 2: Exponentiation

Calculate the power expression, \[3^{2} = 9.\]

Step 3: Simplify the Numerator

Now substitute the values back into the numerator and simplify it: \[6 + 6 + 9 = 21.\]

Step 4: Simplify the Denominator

Evaluate the expression in the denominator, \[18 - 3 = 15.\]

Step 5: Simplify the Fraction

Now, simplify the fraction using the results from the numerator and denominator: \[\frac{21}{15}.\]

Step 6: Reduce the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of 21 and 15, which is 3:\[\frac{21}{15} = \frac{21 \div 3}{15 \div 3} = \frac{7}{5}.\]

Key Concepts

Absolute Value

In mathematics, the absolute value of a number measures how far a number is from zero on the number line. It always results in a non-negative number regardless of whether the input is positive or negative. The absolute value is denoted by vertical bars, like this: \(|x|\). For example, \(|8-2|\) evaluates to \(|6|\), which is simply 6. In real life, absolute value is similar to saying, "I want the distance to zero regardless of direction."

Here's how to understand it:

• Ignore any negative sign in front of a number.
• Calculate the distance as a positive number.

Absolute values are useful in many different scenarios, especially when dealing with differences in mathematical expressions. By converting differences to positive numbers, calculations can be carried out with more predictability.

Exponentiation

Exponentiation involves raising a number to the power of another. It is a way of multiplying a number by itself a certain number of times. In the expression \((3^2)\), 3 is the base and 2 is the exponent, which means we multiply 3 by itself to get 9.

• Base: The number that is going to be multiplied.
• Exponent: The number of times the base is multiplied by itself.

Examples of exponentiation include:

• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(4^2 = 4 \times 4 = 16\)

This concept is commonly used in algebraic expressions as a shorthand notation to simplify computations and representations, making it easier to perform complex calculations.

Fractions

Fractions represent a part of a whole or any number of equal parts. They consist of a numerator and a denominator. In the expression \(\frac{21}{15}\), 21 is the numerator, representing how many parts we have, and 15 is the denominator, showing into how many parts the whole is divided. To simplify a fraction like \(\frac{21}{15}\), you need to find the greatest common divisor (GCD) and divide both the numerator and denominator by it.

Here's how you work with fractions:

• Numerator: Number on top, indicating parts we have.
• Denominator: Number on the bottom, indicating total parts.
• Simplifying: Dividing both parts by their GCD improves readability and calculation.

Understanding fractions and their simplification is crucial in mathematics to make sense of expressions and solve equations accurately.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides exactly without leaving a remainder into two or more numbers. Finding the GCD is essential when simplifying fractions. For instance, in the fraction \(\frac{21}{15}\), we determine the GCD of 21 and 15 to reduce the fraction to its simplest form.

Steps to find the GCD:

• List the factors of each number.
• Identify the largest common factor between them.

In some cases, using the Euclidean algorithm is a quick way to find the GCD. For 21 and 15, the GCD is 3, as 3 divides both 21 and 15 without any remainder. Simplifying using the GCD not only makes fractions more manageable but also facilitates easier computation in subsequent mathematics problems.