Question

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

Short answer

Answer: -5

Step-by-step solution

Calculate the Numerator

Evaluate the numerator by performing the operations in the correct order. Firstly, multiply 3 with -12, and then add 1 to the product: $$3(-12) + 1 = (-36) + 1$$

Simplify the Numerator

Now add -36 and 1 to get the simplified numerator: $$(-36) + 1 = -35$$

Calculate the Denominator

Perform the operation in the denominator. Raise 3 to the power of 2 and then subtract 2 from the result: $$3^2 - 2 = 9 - 2$$

Simplify the Denominator

Now subtract 2 from 9 to get the simplified denominator: $$9 - 2 = 7$$

Simplify the Fraction

Substitute the simplified numerator and denominator values into the fraction: $$\frac{-35}{7}$$

Simplify the Fraction to Its Simplest Form

Dividing -35 by 7 gives us: $$\frac{-35}{7} = -5$$ The simplified form of the given expression is -5.

Key concepts

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This means making the numerator and denominator as small as possible while still keeping the fraction equivalent to the original. Think of it as breaking down a problem into simpler parts.

• Start by identifying the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by the GCD.
• The fraction is now in its simplest form.

For example, in the exercise, the simplified fraction \(\frac{-35}{7}\) is reduced to \(-5\). This means \(35\) and \(7\) share a common factor of \(7\), and dividing both by \(7\) gives a much simpler number to work with. Remember that a fraction like \(\frac{a}{b}\) is simplest when both \(a\) and \(b\) have no common factors other than \(1\).
Reducing fractions makes equations cleaner and easier to handle, especially in more complex algebraic contexts.

Numerator and Denominator

Understanding numerators and denominators is key in tackling fractions. A fraction has two parts: the top number, called the numerator, and the bottom number, known as the denominator.

• The numerator tells us how many parts we have.
• The denominator tells us how many parts the whole is divided into.

For example, in the fraction \(\frac{-35}{7}\), \(-35\) is the numerator, meaning we have \(-35\) parts of something, and \(7\) is the denominator, indicating that the whole is divided into 7 equal parts.
Think of fractions as slices of pizza. If you have \(\frac{3}{4}\) of a pizza, \(3\) is the numerator showing the slices you have, and \(4\) is the denominator showing slices in total. The balance between these two numbers allows you to understand portions easily.

Multiplication and Addition

When simplifying numerical expressions involving both multiplication and addition, it's crucial to follow the correct order of operations - often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). This ensures that calculations are performed correctly.

• First, handle operations inside Parentheses.
• Then compute Exponents.
• Follow with Multiplication and Division, from left to right.
• Finally, perform Addition and Subtraction, from left to right.

In the exercise, we first handled multiplication \(3 \times (-12)\) to get \(-36\). Only then did we add \(1\) to it. By following these steps, we maintained the integrity of the mathematical expression and correctly simplified the numerator.

Exponents

Exponents represent repeated multiplication of a number by itself. For example, \(3^2\) means \(3 \times 3 = 9\). Exponents are a critical part of simplifying mathematical expressions and equations.

• They are often one of the earliest operations to be resolved in an expression.
• Understanding exponents helps manage larger numbers neatly.

In our example, the expression \(3^2 - 2\) required calculating \(3^2\) first, resulting in \(9\), before moving on to subtract \(2\). It’s like packing multiple calculations into a single, compact term.
Mastering exponents means you can efficiently and effectively tackle expressions and equations that might otherwise seem intimidating or complex.