Question

For Exercises \(1-74,\) simplify. $$(-15+12)+5(-6)-[9-(-12)] \div 3$$

Short answer

Question: Simplify the given expression: $$(-15+12)+5(-6)-[9-(-12)] \div 3$$ Answer: -40

Step-by-step solution

Parentheses

First, we need to perform the operations inside the parentheses. There are two sets of parentheses in this expression: \((-15+12)\) and \([9-(-12)]\). For the first parentheses: $$-15+12 = -3$$ For the second parentheses: $$9-(-12) = 9+12 = 21$$ Now our expression becomes: $$-3+5(-6)-(21) \div 3$$

Division

Next, we perform the division operation in the expression. We have one division operation: \((21) \div 3\) $$(21) \div 3 = 7$$ Now our expression becomes: $$-3+5(-6)-7$$

Multiplication

Now, we perform the multiplication operation in the expression. We have one multiplication operation: \(5(-6)\) $$5(-6)=-30$$ Now our expression becomes: $$-3-30-7$$

Subtraction and Addition

Finally, we perform the subtraction and addition operations from left to right. We have two subtraction operations and one addition operation. $$-3-30-7 = (-3-30)-7 = -33-7 = -40$$ The simplified expression is: $$\boxed{-40}$$

Key concepts

Simplifying Expressions

Simplifying expressions is all about making a complex mathematical expression easier to understand and solve. When simplifying, we use a consistent approach that follows specific rules known as the Order of Operations. This systematic method ensures that everyone solves expressions the same way and gets the same result.

The acronym **PEMDAS** stands for:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

In our example, we started by handling parentheses, followed division and multiplication, and finally tackled subtraction. Sticking to this order ensures clarity and correctness.

Parentheses

Parentheses are used to group parts of an expression, indicating that the operations inside should be performed first. They act like brackets in math to prioritize calculations. This is critical in altering how the rest of the operations are executed.

For instance, in the expression \[(-15+12)+5(-6)-[9-(-12)] \] we have:

• First parentheses: \(-15+12\), which simplifies to \(-3\).
• Second group: \([9-(-12)]\) turns into \(9+12 = 21\).

After solving the operations within each set of parentheses, the expression became easier to manage. This step lays the foundation for simplifying the rest of the expression.

Division in Math

Division is about splitting a number into equal parts. It's an operation that reverses multiplication. In expressions, division is processed from left to right, and it follows the operations inside parentheses.

In the provided example, we had \[21 \div 3\]This division simplifies to \[7\]In the simplified expression: \[-3 + 5(-6) - 7\], the division makes the expression cleaner and more straightforward to handle during later steps. It lessens the terms, making the whole expression simpler to work with.

Multiplication in Math

Multiplication represents repeated addition and is a crucial operation performed after dealing with parentheses and division. It combines numbers, making expressions easier to simplify in subsequent steps.

In our example, we encounter multiplication in:\[5(-6)\]This equals \[-30\] when simplified.

Applying the multiplication results, the expression transforms to:\[-3 - 30 - 7\]This outcome aids in preparing for the final adjustments involving subtraction and addition operations. Multiplication, done after division and parentheses, is vital for reducing the expression to its simplest form.