Question

For Exercises \(1-74,\) simplify. $$\frac{6(-4)+16}{(4+7)-9}$$

Short answer

Question: Simplify the expression $$\frac{6(-4) + 16}{4 + 7 - 9}$$. Answer: -4

Step-by-step solution

Calculate the numerator

In the numerator, we have a multiplication and addition operation. We'll start by doing the multiplication first: $$6(-4) + 16 = -24 + 16$$ Now we can do the addition operation: $$-24 + 16 = -8$$ So the simplified numerator is -8.

Calculate the denominator

In the denominator, we have addition and subtraction operations. We'll start by doing the addition operation: $$(4 + 7) - 9 = 11 - 9$$ Now we can do the subtraction operation: $$11 - 9 = 2$$ So the simplified denominator is 2.

Divide the simplified numerator by the simplified denominator

Now that we have simplified the numerator and the denominator, we can divide them to get the final simplified expression: $$\frac{-8}{2} = -4$$ Hence, the simplified expression is -4.

Key concepts

Order of Operations

To ensure mathematical expressions are interpreted consistently, the order of operations is a fundamental concept that dictates the sequence in which arithmetic operations should be performed. The widely accepted order is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This can be easily remembered by the acronym PEMDAS.

When simplifying an expression like \(\frac{6(-4)+16}{(4+7)-9}\), we apply this principle. First, operations inside parentheses are handled, followed by multiplication or division, and finally, addition or subtraction. By following the correct order, we avoid mistakes, and ensure that everyone arrives at the same answer when simplifying expressions.

Arithmetic Operations

The operation symbols in arithmetic are notably '+', '-', '\(*\)', and '\(/\)' for addition, subtraction, multiplication, and division, respectively. When simplifying expressions, it is essential to perform these arithmetic operations with precision. For instance, in the expression \(6(-4) + 16\), we see multiplication and addition. Remember, multiplication has priority over addition.

Multiplying 6 by -4, we get -24, and then adding 16 to -24, we end up with -8. Arithmetic operations are the building blocks of more advanced math, and understanding how to correctly perform them is quintessential for students.

Fraction Simplification

When working with fractions, fraction simplification involves reducing them to their simplest form, making them easier to understand and work with. The process often includes simplifying the numerator and denominator separately before dividing them.

For the given exercise \(\frac{6(-4)+16}{(4+7)-9}\), after applying arithmetic operations accurately, we simplify numerator to -8 and denominator to 2. The last step in fraction simplification is to divide the numerator by the denominator, resulting in -4. Simplifying fractions helps in comparing and calculating with fractions in more complex mathematical contexts.