Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$-3^{2}+2[20 \div(7-11)]$$

Step-by-Step Solution

Answer

The simplified form of the expression \(-3^{2}+2[20 \div(7 - 11)]\) is -1.

Step 1: Evaluate the Exponent

Start with evaluating the expression \(-3^{2}\). In mathematics, the exponent operation is performed before the negation operation. Therefore, it would be \( (-3)^{2} = 9 \).

Step 2: Evaluate the Parentheses

Next, evaluate the expression within the parentheses (7-11). So, it should be \( 7-11 = -4 \).

Step 3: Perform Multiplication and Division

Now perform the division operation \( 20 \div -4 \) which equals to -5. Also, multiply the result by 2 which is \(-5*2 = -10\)

Step 4: Perform Addition

Finally, add the result from step one with the result from step three, which is \(9 + (-10) = -1 \).

Key Concepts

Exponentiation

Exponentiation involves raising a number to the power of another. It's a shortcut for multiplying a number by itself a specific number of times. In this expression, we see \(-3^{2}\).

• First, focus on the base, which is \(-3\).
• The exponent is 2. This means you multiply \(-3\) by itself once.
• Calculate \((-3) \times (-3)\), which equals 9.

So, the result of \((-3)^2\) is 9. Note that exponents come before negative signs unless parentheses indicate otherwise.
Remember, understanding the order here is crucial to avoid mistakes!

Parentheses

Parentheses indicate which operations should be performed first. They help group parts of mathematical expressions, bringing clarity. In our exercise, we must focus on \((7-11)\).

• Calculate the expression inside the parentheses: \(7 - 11\) yields \(-4\).

By doing this, we respect the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Keep the parentheses in mind for clarity and correctness!

Multiplication and Division

Multiplication and division are operations of equal precedence. This means you solve them from left to right as they appear. In this problem, they occur with \(20 \div (-4)\).

• First, perform the division: \(20 \div (-4)\) equals \(-5\).
• Next, the multiplication follows: multiply \(-5\) by 2 to get \(-10\).

By handling multiplication and division this way, you ensure accuracy in your math problems. Keep practicing, and this will become second nature!

Addition and Subtraction

Addition and subtraction are the final steps to complete in the order of operations. They should be applied left to right. Let's look at our problem: we need to add \(9\) and \(-10\).

• Start by looking at your simplified results from earlier steps.
• Next, calculate: \(9 + (-10)\).
• This results in \(-1\).

Remember, adding a negative number is the same as subtracting its positive equivalent. Keeping this in mind helps simplify complex problems!