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Chapter 8: Problem 33

Use a calculator to approximate each square root. Round to three decimal places. $$\sqrt{23}$$

Short Answer

Expert verified
The square root of 23, approximated and rounded to three decimal places, is 4.796.

Step by step solution

01

Identify the Number

Identify the number that you need to find the square root of. In this exercise, the number is 23.
02

Use the Calculator

Use the square root function on your calculator, which is commonly denoted as 'sqrt'. Input 23 into your calculator and press the 'sqrt' button. Make sure to follow the specific instructions for your particular calculator model if you're unsure how to carry out this function.
03

Round the Result

The result that you get will be a number with a series of decimal places. You need to round this number to three decimal places as instructed in the exercise. The rounding rule is that if the digit after the third decimal place is 5 or more, then the number in the third decimal place is increased by 1, else it remains the same.

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Key Concepts

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

Using a Calculator
Calculators are handy tools for quickly finding the square roots of numbers, like 23. Most calculators have a button labeled 'sqrt', which stands for square root.
  • First, identify the number you want to take the square root of; in this case, it's 23.
  • Enter 23 into your calculator.
  • Press the 'sqrt' button.

This process will give you a decimal representation of the square root of 23. It's important to note that each calculator might handle this function slightly differently, so be sure to read your calculator鈥檚 manual if you're unsure of how to perform this calculation. No math problem is too daunting when you have a calculator at your fingertips!
Decimal Approximation
Once you use your calculator to find the square root of 23, it will give you a number with many decimal places. For practical purposes, especially in education, it is common to use a decimal approximation rather than the full decimal value.
Decimal approximation involves using a shorter version of a long decimal to make it easier to read and work with. When a number like \( \sqrt{23} \) is calculated, the approximation is often needed in fields like engineering, physics, and statistics.
  • Approximating decimals can make equations easier to handle.
  • This approach simplifies communication of numerical findings.
  • Always check how many decimal places are required for your task, like three decimal places in this case.

Decimal approximations ensure that you have a manageable number representation without losing significant precision.
Rounding Decimals
Rounding simplifies decimal numbers by keeping them readable while retaining essential accuracy.
To round a number to three decimal places, identify the third decimal place and look at the digit immediately following it. Here are the steps to rounding off:
  • Find the digit in the position you need to round to.
  • Check the digit right after it.
  • If that digit is 5 or more, increase the number in your desired decimal place by 1. Otherwise, leave it as it is.
For example, if the decimal \
representation of \(\sqrt{23}\) results in 4.795831... when rounded to three decimals, it becomes 4.796.
This approach keeps your work neat and the calculations concise. Practice helps in mastering decimal rounding, which is a valuable skill in both academic and everyday tasks.

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Most popular questions from this chapter

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$\left(\frac{1}{4}\right)^{-\frac{1}{2}}$$

$$\text { Simplify: } \sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}}$$

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$32^{-\frac{4}{5}}$$

It is difficult to measure the height of a tall tree, particularly when it is growing in a dense forest. However, it is relatively easy to measure its base diameter. The formula $$h=0.84 d^{\frac{2}{3}}$$ models a tree's height, \(h,\) in meters, in terms of its base diameter, \(d,\) in centimeters. (Source: Thomas McMahon, Scientific American, July, 1975 ) a. The largest known sequoia, the General Sherman in California, has a base diameter of 985 centimeters (about the size of a small house). Use a calculator to approximate the height of the General Sherman to the nearest tenth of a meter. b. Rewrite the formula in radical notation.

Simplify each expression. Write answers in exponential form with positive exponents only. Assume that all variables represent positive real numbers. $$\frac{x^{\frac{1}{6}}}{x^{\frac{5}{6}}}$$
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