Problem 61 - Elementary and Intermediate Algebra

Question

Simplify using a calculator. Round your answer to the nearest thousandth. $$ \frac{2.5^{2}-10 \cdot 12 \div(-1.5)}{(3+5)^{2}-60} $$

Step-by-Step Solution

Answer

21.563

Step 1 - Calculate the numerator

First, calculate each part of the numerator separately. Start by calculating the square of 2.5: \[ 2.5^2 = 6.25 \] Next, calculate the multiplication and division: \[ 10 \times 12 = 120 \] and \[ 120 \div (-1.5) = -80 \] Finally, combine these results: \[ 6.25 - (-80) = 6.25 + 80 = 86.25 \]

Step 2 - Calculate the denominator

Next, calculate each part of the denominator separately. Start by solving the expression inside the parentheses: \[ 3 + 5 = 8 \] Then, calculate the square of 8: \[ 8^2 = 64 \] Finally, subtract 60 from this result: \[ 64 - 60 = 4 \]

Step 3 - Divide the numerator by the denominator

Now, divide the result of the numerator by the denominator: \[ \frac{86.25}{4} = 21.5625 \]

Step 4 - Round the answer to the nearest thousandth

Finally, round the answer to the nearest thousandth: \[ 21.5625 \approx 21.563 \]

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra. It involves reducing equations to their simplest form by performing arithmetic operations step-by-step.
In our example, we start simplification by breaking down the given complex fraction. Calculate all parts separately:

• First, we square the number 2.5 to get 6.25.
• Then, multiply and divide 10 and 12, and divide by -1.5 to get -80.
• Finally, combine these results in the numerator by adding 6.25 and -80.
• Repeat similar steps for the denominator: calculate inside the parentheses and simplify.

Simplifying helps in solving expressions quickly and accurately.

Using Calculators

Calculators are valuable tools for simplifying expressions, especially those involving complicated operations.
When using a calculator for simplification:

• Always follow the order of operations (PEMDAS/BODMAS).
• Enter numbers and operations carefully, checking each step.
• Make use of the parenthesis key to keep parts of the expressions organized.
• Confirm your intermediate results to ensure accuracy.

A calculator helps in avoiding manual mistakes, especially with large and complex numbers, and saves time.

Rounding Numbers

Rounding numbers is the process of adjusting the digits of a number to make it simpler while keeping its value close to the original.
In our exercise, after simplifying the given expression, we get 21.5625.

• To round it to the nearest thousandth, look at the fourth decimal place.
• Since the thousands digit is 2, we keep 21.563 as our rounded number (21.5630).
• Keep practicing with different numbers to get confident in rounding correctly.

Rounding is useful for making numbers easier to interpret and manage.

Order of Operations

Remembering the order of operations is essential when simplifying expressions.
This is often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition, and Subtraction (from left to right)).

• First, solve expressions inside parentheses.
• Then, calculate exponents.
• Next, perform all multiplications and divisions from left to right.
• Lastly, carry out addition and subtraction from left to right.

In our example, following PEMDAS helps in properly solving each part and simplifying the expression correctly.
Always apply the order of operations to avoid errors and ensure accurate results.