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Chapter 13: Problem 97

\(\sqrt{21}\)

Short Answer

Expert verified
\(\sqrt{21} \approx 4.5826\)

Step by step solution

01

Understand the Problem

The task is to find the square root of 21. It means we need to determine a number that, when multiplied by itself, equals 21.
02

Estimate the Square Root

We know that \(\sqrt{16} = 4\) and \(\sqrt{25} = 5\). Since 21 is between 16 and 25, \(\sqrt{21}\) must be between 4 and 5.
03

Refine the Estimate

Try numbers between 4 and 5: 4.5 \(\times\) 4.5 = 20.25 and 4.6 \(\times\) 4.6 = 21.16. Therefore, the \(\sqrt{21}\) is slightly less than 4.6 but more than 4.5.
04

Use a Calculator

To get a more precise answer, use a calculator. The calculator shows \(\sqrt{21}\) is approximately 4.5826 to four decimal places.

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Key Concepts

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

Estimating Square Roots
Estimating square roots is an important skill when you don't have a calculator handy or need a quick approximation.

To start estimating, find the two nearest perfect squares. For \(\sqrt{21}\), we know that 16 and 25 are closest perfect squares since \(\sqrt{16} = 4\) and \(\sqrt{25} = 5\).

Since 21 is between 16 and 25, we know that \(\sqrt{21}\) must be between 4 and 5.
  • \(\sqrt{16} = 4\)
  • \(\sqrt{25} = 5\)
This gives us a good starting point. Next, we'll refine this estimation by checking values between these two numbers.
Refining Estimates
When refining estimates, it's all about narrowing down the range. Statistical data and approximation can help.

Starting with your initial guess between 4 and 5, try squaring values in this range. For \(\sqrt{21}\), we tried 4.5 and 4.6.
  • 4.5 \times 4.5 = 20.25 (too low)
  • 4.6 \times 4.6 = 21.16 (slightly too high)
Since 20.25 is too low and 21.16 is slightly too high, we now know \(\sqrt{21}\) is somewhere between 4.5 and 4.6.

Continuing this method improves accuracy. You could check more values like 4.55, but at some point, it's best to use a tool for precision.
Using Calculators for Precision
For precise calculations, calculators are essential. Continuing from our previous effort, a calculator can quickly provide the exact value of a square root.

To find \(\sqrt{21}\) using a calculator, input 21 and press the square root function. The result, around 4.5826, is accurate to four decimal places.

Calculators are extremely useful for:
  • Complex calculations
  • Checking manual estimates
  • Ensuring accuracy in mathematical work
While estimating is helpful, relying on a calculator for final precision ensures minimal errors.

Combining estimates with calculators gives a well-rounded understanding of square roots, balancing mental math skills and technological aid.

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Most popular questions from this chapter

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 2+\frac{4}{3}+\frac{8}{9}+\cdots $$

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 8\left(\frac{1}{3}\right)^{k-1} $$

Don contributes \(\$ 500\) at the end of each quarter to a tax-sheltered annuity (TSA). What will the value of the TSA be after the 80 th deposit ( 20 years) if the per annum rate of return is assumed to be \(5 \%\) compounded quarterly?

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ \frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1} $$

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 1-\frac{3}{4}+\frac{9}{16}-\frac{27}{64}+\cdots $$
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