Problem 43 - Algebra A Combined Function

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

First, evaluate the absolute value in the numerator. The absolute value of a number is its distance from zero on the number line, without regard to direction. Thus, calculate \(|8-2|\). Since \(8-2 = 6\), we have \(|8-2| = 6\).

Step 2: Calculate the Exponent

Next, evaluate the exponent in the numerator. The exponent indicates that the number is multiplied by itself a given number of times. For \(3^{2}\), it means \(3\times3 = 9\).

Step 3: Substitute Evaluated Terms into the Expression

Substitute the evaluated terms back into the original expression: \(6 + |8-2| + 3^{2}\) to \(6 + 6 + 9\).

Step 4: Simplify the Numerator

Add the numbers in the numerator: \(6 + 6 + 9 = 21\). Now the expression becomes \(\frac{21}{18-3}\).

Step 5: Simplify the Denominator

Simplify the denominator by performing the subtraction: \(18-3 = 15\). Now the expression is \(\frac{21}{15}\).

Step 6: Simplify the Fraction

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

Key Concepts

Absolute Value

Absolute value refers to the distance of a number from zero on a number line, without considering its direction. Basically, it strips the sign from a number, making negative numbers positive, while leaving positive numbers unchanged. In mathematical notation, the absolute value of a number is represented by vertical bars. For example, \(|-5| = 5\) and \(|3| = 3\).
To understand better, imagine a number line. Whether a point is on the negative side or positive side, its absolute value always represents how far it is from zero. Utilizing absolute values is crucial in various mathematical contexts, including equations and inequalities, as it helps establish non-negative measures of distance.
In the example given, the absolute value involved \(8 - 2 = 6\), so \(|8 - 2| = 6\). Hence, the term contributes \(6\) to the expression we are simplifying.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number. When we say \(3^2\), it means \(3 imes 3\), giving us \(9\). The base is \(3\) and the exponent is \(2\), indicating the number of times the base is multiplied by itself. Let's look at another example: \(5^3 = 5 imes 5 imes 5 = 125\).
The utilization of exponents streamlines mathematical expressions, reducing clutter and making calculations more efficient and manageable.
In our expression, simplifying \(3^2\) leads directly to the term \(9\). This result, when evaluated, simplifies the numerator.

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form, where the numerator and the denominator have no common divisors other than \(1\). This involves dividing both by their greatest common divisor (GCD).
Take a fraction such as \(\frac{14}{28}\): both \(14\) and \(28\) can be divided by \(14\), yielding \(\frac{1}{2}\).

• It’s important as it allows us to represent quantities more clearly, making them easier to interpret and use in further calculations.
• Simplification makes comparisons between fractions more straightforward.

For the expression in question, \(\frac{21}{15}\) was simplified by dividing top and bottom by their \(GCD = 3\), resulting in \(\frac{7}{5}\).

Greatest Common Divisor

The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both without leaving a remainder. Finding the GCD is pivotal for fraction simplification.
To find the GCD:

• List the factors of each number. For instance, the factors of \(12\) are \(1, 2, 3, 4, 6,\ and \12\).
• Identify the largest factor common to each list.

Utilize it by dividing both the numerator and denominator of a fraction, streamlining the values, as seen with \(\frac{21}{15}\), where the GCD is \(3\). Division by \(3\) gives the simplest form \(\frac{7}{5}\), ensuring the fraction is concise and easily applicable in further math operations.