Problem 37 - Elementary and Intermediate Algebra

Question

Simplify. $$ 3(-10)^{2}-8 \div 2^{2} $$

Step-by-Step Solution

Answer

298

Step 1: Simplify the Expression Inside the Parentheses

Start by simplifying the expression inside the parentheses. For \( (-10)^2 \), square the number -10.

Step 2: Calculate the Square

Calculating \( (-10)^2 \) gives us \( 100 \).

Step 3: Multiply by 3

Multiply the result by 3. So, \( 3 \times 100 = 300 \).

Step 4: Simplify the Division Expression

Next, simplify \( 8 \div 2^2 \). First, calculate \( 2^2 = 4 \).

Step 5: Perform the Division

Now, divide 8 by 4. \( 8 \div 4 = 2 \).

Step 6: Combine the Results

Combine the results of the multiplication and division expressions. \( 300 - 2 = 298 \).

Key Concepts

Order of Operations

When simplifying algebraic expressions, it is crucial to follow the correct sequence of operations. This sequence is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
If you do not follow this order, you may end up with the wrong answer.
For example, in the expression given, we must first address the parentheses and exponents before moving on to multiplication and division.

Exponents

Exponents represent repeated multiplication of a base number. In the exercise expression \( 3(-10)^2 - 8 \div 2^2 \), the part \((-10)^2\) means that we square -10.
This gives us:

• \((-10) \times (-10) = 100\)
  Always remember that the square of a negative number results in a positive number.

Next, in the expression, we also have \(2^2\):

• \(2^2\) simply means multiplying 2 by itself, which equals 4.
This is crucial before proceeding with the division part of the expression.

Multiplication and Division within Algebraic Expressions

Multiplication and division should be handled from left to right following the order of operations.
In the given expression \(3(-10)^{2}-8 \div 2^{2}\), after handling the exponents, we move on to multiplication and division.

• First, compute the multiplication: \(3 \times 100 = 300\).
• Next, simplify the division part: \(8 \div 4 = 2\).

Finally, combine these results: \(300 - 2\).
Computing this gives you 298, which is the simplified form of the given algebraic expression.