Question

\- \(\sqrt{129}\) is between which numbers? (A) 10 and 11 (B) 11 and 12 (C) 14 and 15 (D) 64 and 65

Short answer

The square root of 129 lies between the square root of the nearest perfect squares. Since \(11^2 = 121\) and \(12^2 = 144\), we have \(11 < \sqrt{129} < 12\). Therefore, the correct answer is (B) 11 and 12.

Step-by-step solution

Understand the concept of square root and perfect squares

The square root of a number is a value that, when it is multiplied by itself, gives the original number. A perfect square refers to a number that can be expressed as the product of some integer with itself.

Calculate the square root of the nearest perfect squares

We can start by finding the perfect squares that are closest to 129. Considering that \(12^2=144\) and \(11^2=121\), we can determine that 129 lies between these two perfect squares.

Check which interval √129 belongs to

As we established before, \(12^2 = 144\) and \(11^2 = 121\). The square root of 129 falls between these two perfect squares, so we can write it as \(11 < \sqrt{129} < 12\).

Choose the correct answer

We have found the interval for the square root of 129. Comparing it with the given options, we can conclude that the correct answer is (B) 11 and 12.

Key concepts

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. For example, numbers like 1, 4, 9, 16, 25, and so on are perfect squares. They are simply formed by multiplying a whole number by itself, such as 3 \(\times\) 3 = 9 or 5 \(\times\) 5 = 25. Recognizing perfect squares is crucial when working with square roots because they help us estimate or find the exact values effortlessly.
For instance, if you're trying to find the square root of 144, you know it's a perfect square because 12 \(\times\) 12 = 144. Understanding these numbers provides a foundation for estimating more complex square roots that don't resolve to whole numbers easily.

Estimating Square Roots

Estimating square roots is all about finding which two perfect squares a given number falls between. For instance, if you want to estimate \(\sqrt{129}\), you would look for perfect squares close to 129.

• Identify nearby perfect squares. For 129, these are 121 \((11^2)\) and 144 \((12^2)\).
• Recognize that the number 129 lies between these two, meaning \(\sqrt{129}\) is also between 11 and 12.

There might be no quick answer for the exact square root of numbers like 129 without using a calculator. However, by using the proximity to perfect squares, you can infer that \(\sqrt{129}\) is approximately 11.3 or so. This method provides a practical way for quick approximations, especially useful for mental math and rough calculations.

Number Sense

Number sense is the intuitive understanding of numbers, their magnitude, and the relationships between them. It's crucial when estimating square roots as it involves recognizing patterns and approximations. Developing strong number sense means being able to make educated guesses and approximations quickly. For example, knowing immediately that \(\sqrt{50}\) is slightly more than 7 because it falls between 49 \((7^2)\) and 64 \((8^2)\). With practice, you can hone this skill:

• Regularly test yourself with random numbers to estimate their square roots.
• Try approximating numbers quickly in your head before verifying with a calculator.

A strong number sense helps simplify complex math problems, making them more approachable and less intimidating. It allows students to become more confident in their math skills beyond the classroom.