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Chapter 0: Problem 143

Use a calculator to approximate the expression to 2 decimal places. $$\frac{-3+5 \sqrt{2}}{7}$$

Short Answer

Expert verified
0.58

Step by step solution

01

- Understand the expression

The given expression is \(\frac{-3+5 \sqrt{2}}{7}\). It involves a combination of regular arithmetic and a square root.
02

- Compute the square root

Use a calculator to find the approximate value of \(\sqrt{2}\), which is approximately 1.414.
03

- Substitute and simplify

Replace \(\sqrt{2}\) with 1.414 in the expression: \(\frac{-3 + 5 \times 1.414}{7}\). This simplifies to \(\frac{-3 + 7.07}{7}\).
04

- Combine terms in the numerator

Add the terms in the numerator: -3 + 7.07, which equals 4.07.
05

- Divide the result by 7

Finally, divide 4.07 by 7 to get approximately 0.58.

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Key Concepts

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

square root approximation
Square root approximation is about finding an estimated value for a square root when an exact value is difficult or impossible to calculate. In our exercise, we need to calculate \(\frac{-3+5 \sqrt{2}}{7}\). \(\sqrt{2}\) is an irrational number meaning it has infinite non-repeating decimals.

However, for our purposes, approximating \(\sqrt{2}\) to two decimal places provides us with a value of 1.41.
Hence, we use the approximate value 1.41 in the expression to simplify our calculations.
This makes it easier to handle the arithmetic operations with a manageable number.
arithmetic operations
Arithmetic operations are the basic calculations including addition, subtraction, multiplication, and division. Let's break down the given expression to understand its arithmetic operations:
  • The original expression is \(\frac{-3+5 \sqrt{2}}{7}\).
  • Substitute \(\sqrt{2}\) with its approximate value, 1.41, to get the expression \(\frac{-3+5 \times 1.41}{7}\).
  • First, perform the multiplication within the numerator: 5 \times 1.41 = 7.07.
  • Next, carry out the addition in the numerator: -3 + 7.07, which simplifies to 4.07.
  • Finally, divide the result by 7: \(\frac{4.07}{7}\), which approximately equals 0.58.
By breaking the expression into smaller, more manageable steps, it becomes easier to follow and understand each arithmetic operation.
using a calculator
Using a calculator helps us to quickly and accurately perform arithmetic operations, especially with complex numbers like square roots. Here's how you can efficiently use a calculator for our expression: \(\frac{-3+5 \sqrt{2}}{7}\):
  • First, find the square root of 2 using the calculator, which gives you approximately 1.41.
  • Next, multiply 5 by the result (1.41) to get 7.07.
  • Add -3 to 7.07 to obtain 4.07.
  • Finally, divide 4.07 by 7 to get approximately 0.58.
Each step involves simple arithmetic operations made easier and more precise with the help of a calculator.
Ensure to follow the order of operations (PEMDAS/BODMAS) to maintain accuracy in your calculations.
The calculator assists in reducing human errors and provides a quicker solution, enhancing your efficiency in problem-solving.

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