Problem 97 - Algebra and Trigonometry

Question

\(\sqrt{21}\)

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

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.