Question

For Exercises \(1-74,\) simplify. $$\frac{20+12(-3)}{3^{2}-1}$$

Short answer

Answer: -2

Step-by-step solution

Simplify the numerator

In the numerator, we have the expression \(20 + 12(-3)\). First, we need to multiply \(12\) with \(-3\). $$12 \times -3 = -36$$ Now, we add this result to \(20\). $$20 - 36 = -16$$ So, the numerator simplifies to \(-16\).

Simplify the denominator

In the denominator, we have the expression \(3^2 - 1\). First, we need to evaluate the exponent \(3^2\). $$3^2 = 9$$ Now, we subtract \(1\) from this result. $$9 - 1 = 8$$ So, the denominator simplifies to \(8\).

Divide the simplified numerator by the simplified denominator

Now that we have simplified the numerator and denominator, we need to find the quotient. $$\frac{-16}{8} = -2$$ So, the simplified expression is \(-2\).

Key concepts

Understanding Numerical Expressions

Numerical expressions are mathematical phrases that include numbers and operations, but no equal sign. An example from our problem is \( 20 + 12(-3) \). These expressions often require simplification to a single numerical value.

• They include operations like addition, subtraction, multiplication, and division.
• Expressions can also contain exponents or powers, which are repeated multiplications.

To simplify a numerical expression, all operations within it need to be performed. The goal is to reduce the expression until only one number remains.

Mastering the Order of Operations

The order of operations is a set of rules that indicate the sequence in which operations should be performed to ensure consistency. Using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) can help remember the order.
In our problem, after determining that the numerator is \( 20 + 12(-3) \), we recognize that multiplication comes before addition.

• First, carry out the multiplication: \(12 \times -3\).
• After simplifying that part of the expression, proceed to the addition.

Following the correct order is essential to accurately compute the expression's value.

Performing Division

Division is the process of splitting a number into equal parts. In our exercise, we have reached the point of dividing the simplified numerator by the simplified denominator: \( \frac{-16}{8} \).

• The result of dividing \(-16\) by \(8\) is \(-2\).
• This operation determines how many times the number in the denominator fits into the numerator.

In general, division helps simplify complex fractions to find a single number as the answer. Accuracy in division is crucial for the correct final value of any expression.

Tackling Multiplication in Expressions

Multiplication involves determining the total of adding a number to itself multiple times. In the given exercise, vital multiplication occurs when solving the numerator: \(12 \times -3\).

• This step alters the sign of the result due to the involvement of a negative number.
• The product calculated is \(-36\) because the multiplication of a positive and a negative results in a negative.

Understanding multiplication as a concept of repeated addition helps when numbers are large or include negative numbers.

Working with Exponents

Exponents represent the number of times a base number is multiplied by itself. In our expression's denominator, \(3^2 - 1\), the exponent \(2\) indicates that the base \(3\) multiplies by itself.

• \(3 \times 3 = 9\), illustrating the process of using exponents.
• This simplifies the denominator's expression, allowing further subtraction.

Exponents greatly reduce the length of expressions by representing repeated multiplication succinctly. They are efficient and simplify calculations in expressions.