Square Root of 784

The square root of 784 is a number whose square is 784. A perfect square is a number that can be expressed as a square of any integer, 784 is a perfect square number as (± 28)² = 784. The square of 784 in radical form is written as √784 and in the exponential form, it is written as (784)^½. In this article, we shall learn how to calculate the square root of 784 using the prime factorization and long division method.

Learn more about square and square roots.

What is the Square Root of 784?

The square root of 784 is a number that is when multiplied twice by itself, gives the answer 784. The number 784 is a perfect square as we can find integers 28 and –28, whose square is 784.

Now, we have the quadratic equation x² – 784 = 0, on solving the equation,

x² – 784 = 0

⇒ x² = 784

Taking square root on both sides, we get

⇒ x = √784

⇒ x = ± 28

Thus, roots of the above quadratic equation are the square root of 784.

Also read: Properties of square numbers.

How to Find the Square Root of 784?

Now, we shall find the square root of 784 using the prime factorisation and long division method.

Repeated Subtraction Method

Square of any perfect square number can be determined by the repeated subtraction method. In this method, we have to repeatedly subtract the given number by consecutive odd numbers until we get the answer zero. The nth odd number for which we get the answer zero is the square root of the given number.

Thus, we have 784 – 1 = 783, 783 – 3 = 780, 780 – 5 = 775, continuing in this way we get for the 28th odd number the answer is zero.

∴ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31 + 33 + 35 + 37 + 39 + 41 + 43 + 45 + 47 + 49 + 51 + 53 + 55 + = 441.

For larger numbers, it is not very efficient to use this method. Instead, we can use the prime factorisation method to calculate the square roots.

Prime Factorisation Method

To find the square root of 784 by the prime factorisation method, we first prime factorise 784 and then make pairs of two to get the square root.

Prime factorisation of 784 = 2 × 2 × 2 × 2 × 7 × 7

The square root of 784 = √[2 × 2 × 2 × 2 × 7 × 7] = 2 × 2 × 7.

Long Division Method

To find the square root of 784 by the long division method, we shall write 784 as the dividend and pair its digits from right to left. Now, we shall calculate the square root of 784 as follows:

To learn how to find the square root of any number long division method, click here.

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Solved Examples on Square Root of 784

Example 1:

Find the length and breadth of a rectangular plot whose area is 1568 m² and length is twice its breadth.

Solution:

Let ‘b’ be the breadth of the rectangular plot

Length of the rectangular plot = 2b.

Area of the rectangular plot = length × breadth = 1568 m²

⇒ 2b × b = 1568

⇒ b² = 1568/2 = 784

⇒ b = √784 = 28 m (taking positive root as length cannot be negative)

∴ the length of the rectangular plot = 2 × 28 = 56 m

And the breadth of the rectangular plot= 28 m.

Example 2:

Find the roots of the equation (x² – 500)/2 = 142.

Solution:

We have, (x² – 500)/2 = 142

⇒ x² – 500 = 284

⇒ x² = 284 + 500 = 784

Taking square root on both side of the equation, we get

⇒ x = √784

⇒ x = ± 28

Example 3:

Find the diameter of a circle whose area is 784𝜋 cm².

Solution:

Let r be the radius of the circle.

Area of the circle = 𝜋r² = 784𝜋 cm²

⇒ r² = 784

⇒ r = √784 = 28 (taking positive root as length cannot be negative).

∴ the diameter of the circle = 2 × 28 = 56 cm.