Estimating Square Root and Cube Root

Finding square root and cube root of a number is an integral part of studying science, whether it is in Chemistry, Physics or Maths itself. But if we observe carefully, then squares are quite easier than finding square roots and gets more intense if we have to take out the cube root. Since they are quite important for daily use if you are involved in technical fields, therefore we use some brilliant tricks to hack the game of square and cube roots. These tricks can help you estimate the numbers quicker than ever and will also help you solve equations easily. 

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.

FAQ (Frequently Asked Questions)

Question: How to Calculate the Square Root of Negative Numbers?

Solution: Square of any number is always positive because whenever a negative number is multiplied with another negative number, it results in a positive value. Since the square root of a number gives us the number’s value when multiplied by itself, it is not feasible for any negative number to be multiplied with its own; hence the real roots of the number do not exist. Instead, a new concept of imaginary numbers is introduced, which deals with the square root of negative numbers and is represented by the symbol iota (i).

Question: How to Find the Cube Root of a Negative Number?

Solution: To find the cube root, we need to proceed in the traditional way. When any negative number is multiplied by itself, it turns positive, but if it is again multiplied by itself, then it turns again into a negative number because negative multiplied by a positive number is negative. Hence, unlike the square root, the cube root can be negative despite being real in nature. The only thing to consider here is the fact that the cube root of any number is always negative.

Question: How to Estimate Values of Numbers with Decimal Points?

Solution: To estimate the cube and square roots of the number, we need to start classifying away from the decimal point. That is we need to take the doubles and triplets from the left of decimal for the whole numbers part and for the decimal part we need to take the bars from the decimal towards the left and continue until the decimals exhaust. In decimal numbers the number of bars decide the number of digits after the decimal in the root value.