Problem 64 - Fundamentals of Mathematics

Question

Find the roots (using your knowledge of multiplication). Use a calculator to check each result. \(\sqrt{121}\)

Step-by-Step Solution

Answer

The square root of 121 is 11.

Step 1: Understand the Problem

We need to find the square root of 121, which means we are looking for a number that, when multiplied by itself, equals 121.

Step 2: Estimate

Think of perfect squares close to 121. For example, 100 is a perfect square with a root of 10, and 144 is a perfect square with a root of 12. The square root of 121 should therefore be a whole number between these two.

Step 3: Test Multiplication

Calculate the square of 11: \(11 \times 11 = 121\). This shows 11 squared results in 121, confirming that 11 is the square root.

Step 4: Verify with a Calculator

Use a calculator to find the square root of 121. Enter \(\sqrt{121}\) and confirm the result is 11.

Key Concepts

Perfect squares

Understanding perfect squares is key to solving many square root problems. A perfect square is a number that can be expressed as the product of an integer with itself. Learning to recognize these numbers can make finding square roots much easier. For example:

• The number 4 is a perfect square, because it equals \(2 \times 2\).
• The number 9 is another perfect square, as it equals \(3 \times 3\).
• The number 121 is also a perfect square, because it is \(11 \times 11\).

Recognizing the patterns in these numbers helps in mental math and estimation.
Understanding perfect squares also allows you to quickly deduce the square root of a number when it is a perfect square itself, as is the case with 121.

Estimation

Estimating square roots is a valuable skill, especially when dealing with numbers that are not perfect squares. The goal of estimation is to quickly determine an approximate value that is rational and easy to work with.

• First, identify the perfect squares closest to the number under the square root. For 121, these are 100 (with a square root of 10) and 144 (with a square root of 12).
• Recognize where the number stands in relation to these perfect squares. Since 121 lies perfectly between 100 and 144, you can infer that its square root will be between 10 and 12, and more accurately, it's likely to be a whole number right in between - hence 11.

Such estimation techniques simplify the process when dealing with larger, more complex numbers. They also provide a good initial guess before detailed calculation or verification.

Verification with a calculator

While manual calculation is important, using a calculator provides a quick way to verify your answers. This is especially useful in confirming complex calculations or when precision is required.
To verify a square root using a calculator:

• Step 1: Turn on your calculator and locate the square root button (often labeled as \(\sqrt{}\)).
• Step 2: Enter the number you want the square root of. For example, type 121.
• Step 3: Press the square root button to compute the square root.
• Step 4: Check if the result matches your manual calculation. For \(\sqrt{121}\), the calculator should display 11, confirming your earlier steps.

Using a calculator is also a great way to handle larger numbers or decimals where manual calculation becomes tedious. Despite this, relying on both methods ensures you have a thorough understanding of the concept.