Problem 42 - College Algebra

Question

For the following exercises, simplify the expression. $$ 5 l \div 3 l \cdot(9-6) $$

Step-by-Step Solution

Answer

The expression simplifies to 5.

Step 1: Solve inside the parenthesis

The expression contains a parenthesis that we will solve first. Inside the parenthesis, we have the calculation \(9 - 6\). Subtract 6 from 9 to get: \(9 - 6 = 3\).

Step 2: Replace and simplify

Replace the original expression with the simplified parenthesis: \(5l \div 3l \cdot 3\).

Step 3: Divide the terms involving 'l'

Perform the division \(5l \div 3l\). \(l\) cancels out and we are left with \(\frac{5}{3}\) since any non-zero value divided by itself equals 1: \(\frac{5l}{3l} = \frac{5}{3}\).

Step 4: Multiply the remaining numbers

Now, multiply the result from the division \(\frac{5}{3}\) by the result of the parenthesis which is 3. Thus, \(\frac{5}{3} \cdot 3 = 5\).

Key Concepts

Order of Operations

When simplifying algebraic expressions, it's important to follow a specific order to ensure accuracy. This is known as the "order of operations." A common acronym to remember this is PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Let's break it down:

• Parentheses: Always start with operations inside parentheses. This means you evaluate expressions enclosed in brackets first.
• Exponents: If there are any exponents, they are handled next. However, in our example, there aren't any.
• Multiplication and Division: These are equal priority operations. You perform them from left to right as they appear in the expression.
• Addition and Subtraction: Lastly, handle addition and subtraction, again working from left to right.

Applying these rules to our exercise, we first calculate inside the parentheses: \(9 - 6\), resulting in 3. Then, according to the order of operations, we handle the division and multiplication next. Following this order ensures correct simplification of the expression.

Division in Algebra

In algebra, division works similarly to regular division, but you must be careful with variables and terms. When dividing two algebraic terms, you simplify by dividing the coefficients and canceling out any common variables.In our example, we start with the division \(5l \div 3l\). Here, you notice the variable \(l\) appears in both the numerator and the denominator. This allows us to cancel \(l\) out, simplifying the expression to \(\frac{5}{3}\). Remember, division by an expression equals \(1\) if the expression itself is not zero.Important points to consider:

• If both the numerator and denominator contain the same variable, they can be canceled out.
• After cancelation, continue with the coefficients (numbers in front of the variables).
• Always ensure that your division is valid by checking the denominator isn't zero.

Multiplication in Algebra

Multiplication in algebra often involves coefficients and variables. Each part needs to be handled individually, paying attention to the rules of multiplication.After handling division in our expression, we proceed to multiply \(\frac{5}{3}\) by the number obtained from the parentheses, which is 3. This is straightforward:

• Multiply the coefficient \(\frac{5}{3}\) by 3, giving \(\frac{5}{3} \cdot 3\).
• To simplify, realize \(\frac{5}{3} \cdot 3 = 5\) since \(3 \times \frac{1}{3} = 1\).

Multiplication tips:

• When multiplying fractions, multiply the numerators together and the denominators together.
• If a number is present (like 3 in this example), it's treated as \(\frac{3}{1}\).
• Multiply coefficients separately if variables are involved, keeping the laws of exponents in mind for any variables.

By following these procedures, multiplication in algebraic expressions becomes manageable and clear.