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Chapter 6: Problem 10

Simplify. $$7.5+2.2 \sqrt{0.25}-36.8 \div 8$$

Short Answer

Expert verified
Question: Simplify the expression: \(7.5 + 2.2 \times \sqrt{0.25} - 36.8 \div 8\). Answer: The simplified expression is \(4\).

Step by step solution

01

Evaluate the square root

Since \(\sqrt{0.25} = 0.5\), the expression becomes: $$7.5 + 2.2\times 0.5 - 36.8 \div 8$$
02

Perform the multiplication

Now, we need to perform the multiplication \(2.2\times 0.5 = 1.1\): $$7.5 + 1.1 - 36.8 \div 8$$
03

Perform the division

Next, we have to perform the division \(36.8\div 8=4.6\): $$7.5 + 1.1 - 4.6$$
04

Simplify the expression

Finally, simplify the expression by adding and subtracting the numbers: $$7.5 + 1.1 - 4.6 = 8.6 - 4.6 = 4$$ So, the simplified expression is: \(4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplification Process
Simplifying an arithmetic expression means breaking it down into simpler parts. When faced with an expression, identify operations that can be performed step by step. This approach prevents errors and ensures you handle complex calculations correctly.

Start by isolating smaller operations within the expression. Evaluate components like square roots and multiplications first, as they'll impact the entire expression. Once these are simplified, you can tackle addition or subtraction. The process is about reducing the complexity stage by stage until the final answer is clear and simple.
Square Roots
A square root is a number that, when multiplied by itself, gives the original number. In our exercise, you see this with \(\sqrt{0.25}\), where the goal is to find a number that squared makes \(0.25\).

It's like finding the sides of a square knowing the area. For \(0.25\), the square root is \(0.5\), because \(0.5 \times 0.5 = 0.25\). Identifying square roots simplifies expressions by replacing the root with its numerical value, making the equation easier to solve.
Multiplication and Division
These operations often go hand in hand and impact expressions significantly. In our problem, we multiply \(2.2 \ imes 0.5\), resulting from the square root simplification, leading to \(1.1\).

After that, handle divisions, such as \(36.8 \div 8\), which simplifies this part of the expression to \(4.6\).
  • Always perform multiplication and division before addition and subtraction, according to the order of operations.
  • Breaking down these operations step-by-step ensures a correct simplification.
These calculations help in reducing the expression to simpler numbers for further simplification.
Order of Operations
The order of operations is essential when simplifying expressions. It provides a standard way to solve mathematical problems. You might know it as PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction.

In our exercise, you first tackle the square root and follow with multiplication and division before handling addition and subtraction. This order prevents mistakes and ensures the expression is reduced correctly.
  • Start with operations inside any parentheses.
  • Handle exponents and square roots next.
  • Proceed with multiplication and division.
  • Finally, perform addition and subtraction.
Following these rules ensures clarity and precision in arithmetic expressions.

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