Square Roots
When two numbers are multiplied with each other, they form the square of the number. It is named square because if you feed the dimension in one side of the square, then the area of the corresponding square will give you the square of a number. When we break down these squares to their primitive numbers, then the resulting number is called a square root of the base number.
The traditional method of finding square roots is by prime factorisation the number and then pairing the same divisors into a single group and considering them as one; then they are all multiplied to give us the square root of the base number. Some square root examples:
\[\sqrt[2]{36}\] = 6
\[\sqrt[2]{81}\] = 9
Cube Roots
When three numbers are multiplied with itself, then the resulting figure is termed as the cube of the number. It is derived from the fact that, if the sides of a cube are taken as the number respectively then the resulting output is the volume of the cube. When this cube of a number is broken down to its primitive form, then the numbers so obtained are known as the cube root of the base number. The traditional method to find the cube root of a number is by prime factorising the cube number and then arranging the divisor into a group of three same numbers and then assuming them as a single entity. After that, all the individual pieces are multiplied, which gives us the output as a cube root. Some cube root examples:
\[\sqrt[3]{1728}\] = 12
\[\sqrt[3]{343}\] = 7
Estimating Square Root
The process of estimating square roots is quite easy; let’s approach it step by step. The first step to determine the root of a number is to arrange it into doublets from the right. After that, the numbers of doublets determine the number of digits in the root. After that, the nearest square is considered and from both directions, i.e. the largest perfect square near it and the smallest square after it. Then the square is estimated accordingly
Let’s understand it with an example.
For the estimation, we take the number 326.
Now we choose dual numbers from the right side, i.e. 3 26 since it has two bars; hence the root will have two numbers.
Now, we will estimate the nearest square root. We know that the square of the number 18 is 324 and the square of the number 19 is 361. Hence the number lies between 18 and 19.
If we observe carefully then the square of 18 is nearer to the number than the square of number 19; hence we can estimate the square root as 18.
Estimating Cube Root
For estimation of cube root, we need to follow the similar steps to that of estimating the square, but now instead of taking two numbers at a time, we will take three and then proceed by comparing the nearest cubes of the number.
Let’s try it with an example.
For the estimation process, we take the number 13824.
Now we break it into groups of three 13 824.
The last digit of the first triplet gives us the number 4, which means that the one’s place in our cube root is 4 itself because only the cube of 4 ends with 4 since the number is a two-digit number because of two bars we need to figure out the tens place next.
The number 13 is closest to the cube of 2 that is 8, and the next number closest is 3, which gives us 27 since we consider the lowest limit; hence the tens digit is 2.
Therefore we get the cube root as 3.
FAQs on Estimating Square Root and Cube Root
1. What does it mean to estimate a square root or a cube root?
Estimating a root means finding an approximate value for the square root or cube root of a number that is not a perfect square or perfect cube. It is a quick method to find a number that is close enough to the actual root without performing complex calculations. For example, to estimate the square root of 27, we know it lies between the square root of 25 (which is 5) and the square root of 36 (which is 6), so the answer is a little more than 5.
2. How can you estimate the square root of a non-perfect square number?
To estimate the square root of a non-perfect square, you can use the bracketing method. Here are the steps:
- Identify the two perfect squares closest to your number, one smaller and one larger.
- The square root of your number will lie between the square roots of these two perfect squares.
- Determine which perfect square your number is closer to. This will give you a more accurate estimate. For instance, to estimate √85, it is between √81 (9) and √100 (10). Since 85 is closer to 81, the estimated value is slightly more than 9, perhaps 9.2.
3. What is the estimation method for finding the cube root of a large number?
The estimation method for finding the cube root of a large perfect cube involves grouping the digits. Follow these steps as per the CBSE Class 8 syllabus for 2025-26:
- Step 1: Start from the right and group the number's digits into sets of three.
- Step 2: Look at the last group (on the right). The unit digit of this group determines the unit digit of the cube root. For example, if the number ends in 8, its cube root will end in 2.
- Step 3: Look at the first group (on the left). Find the largest cube that is less than or equal to this number. The cube root of this number will be the tens digit of your answer.
- Step 4: Combine the tens and unit digits to get the estimated cube root.
4. Why is estimating square and cube roots a useful skill in real life?
Estimating roots is a practical mental math skill with several real-world applications. It is important for:
- Quick calculations: When you don't need an exact answer, like figuring out the approximate side length of a square-shaped garden of 150 sq. ft.
- Problem-solving: In science and engineering, estimation helps in checking the feasibility of a result without a calculator.
- Building number sense: It strengthens your understanding of the relationships between numbers and their powers, which is a foundational concept in mathematics.
5. What is the main difference between finding the cube root of a negative number and the square root of a negative number?
The main difference lies in whether a real number solution exists. A negative number has a real cube root because a negative number multiplied by itself three times results in a negative number (e.g., -4 x -4 x -4 = -64). Therefore, the cube root of -64 is -4. However, a negative number does not have a real square root because any real number (positive or negative) multiplied by itself always results in a positive number. The square root of a negative number, like √-16, is an imaginary number.
6. How do you estimate the square root of a decimal number?
To estimate the square root of a decimal number, like in the long division method, you pair the digits. Start at the decimal point and make pairs of digits going outwards. For the whole number part, you pair from right to left. For the decimal part, you pair from left to right, adding a zero if needed to complete a pair. The number of pairs after the decimal point in the number will determine the number of decimal places in its square root.
7. How does knowing the unit digit of a perfect cube help in its estimation?
Knowing the unit digit of a perfect cube is a powerful shortcut in estimation because each digit from 0 to 9 has a unique unit digit for its cube. For example:
- A number ending in 1 will have a cube root ending in 1.
- A number ending in 8 will have a cube root ending in 2.
- A number ending in 7 will have a cube root ending in 3.
This one-to-one relationship allows you to instantly determine the unit digit of the cube root just by looking at the unit digit of the original number, making the first step of the estimation process very fast and accurate.
