What Is the Square Root of 72?

Mathematics 936745 1280 Credit: Geralt/Pixabay

A perfect square is a number with an integer as its square root. This means that it's a product of an integer with itself. In decimal representation, the square root of 72 is 8.485 when rounded to four significant figures. Because it's not a square number or perfect square, you can solve it quickly with a scientific calculator. If this option is unavailable, there are a couple of ways you can use to arrive at a solution. If this alternative can't be used, there are a couple of tricks you can use to arrive at a solution.

Multiplying Using Two Reference Numbers

To find the square root of 72, you'll need to know the two closest perfect squares surrounding the number 72. In this case, 8 and 9 are our numbers. The number 72 falls between the squares of 8 and 9, which are 64 and 81, respectively.

Thereafter, divide 72 by 8 or 9. The result will be 9 or 8, depending on the number chosen to divide it with. Next, find the mean of that result and the original square used, which is (9+8)/2=8.5. Finally, repeat the previous two steps until you attain the desired accuracy. 

Simplified Radical Form

Expressing in simplified radical form entails simplifying a radical till no more square roots, cube roots, fourth roots, and so on can be found. It also involves the removal of radicals in the denominator of a fraction. The square root of 72 can be simplified by breaking the radicand up into a product of known factors. Begin by finding the highest square that divides into 72 evenly. In this case, 36 is the number. Thus, 72 can be expressed as 36 x 2, then proceed as follows:

√ 72 = √2x√36 = √2x√62 = 6√ 2

Cross Multiplication

The method is used to calculate the exact answer for a square root. For even numbers of digits, you need to multiply the first digit by the last digit, the second by the second last, and so on, until all the digits have been multiplied. Find the sum and double the total. Use the same process for an odd number of digits till you reach the middle digit. Find the sum of the answers and double the total. Next, square the middle digit and add it to the total.

Long Division Method

Start by grouping the digits into pairs, beginning with the units digit. The pair formed and remaining digit (if any) is called a period. After establishing the largest square number equal to or slightly less than the first period, use it as both the divisor and the quotient. Multiply the divisor and quotient then subtract it from the first period. Then, write the next period to the right of the remainder. As a result, a new dividend is formed. This becomes the new dividend.

Now, the new divisor is found by using two times the quotient and adding with it an ideal digit that acts as the next digit of the quotient. The number is chosen, such as the product of the digit, and the new divisor's value equal to or slightly less than the new dividend. Finally, repeat the second, third, and fourth steps until all the periods have been incorporated or taken up. The quotient obtained is the required square root of the given number.