# Find Square Root of 35 by Long Division & Newton Raphson Method (2023)

Values of square root of 35 in the exponent form is $$\sqrt 35$$ or $$35^{0.5}$$ and if we round off the value of square root of 35 to 6 decimal places, we get 5.916079. It is an irrational number. Square root of a number $$a$$ is a value $$b$$ such that, $$a = b \times b$$. Square root symbol is $$\sqrt{}$$. You can estimate the square root by finding the number that, when multiplied by itself, is closest to the target number.

## What is the Square Root of 35?

Square root of 52 is 5.916079 because $$5.916079 … \times 5.916079 \approx 52$$. It is an irrational number which means that it cannot be expressed exactly as the ratio of two integers in the form $$p/q$$.

Square root of 52 is not a rational number, it is an irrational number, and it cannot be simplified as the ratio of two integers. An approximate value for the square root of 52 is 5.916079 but it is a non-repeating and non-terminating decimal and cannot be written as a finite decimal or a fraction.

## How to Calculate Square Root of 35?

Square root of 35 can be found by the various methods which are given below:

• Square Root of 35 by Newton raphson
• Square Root of 35 by Babylonian Method or Hero’s Method
• Square Root of 35 by Long Division Method

## Square Root of 35 by Long Division Method

Here we will discuss how to calculate the square root of $$35$$ using the long division method. This method is the lost art of how to calculate the square root of $$35$$ by hand before modern technology was invented. Follow the below steps to find the square root of $$35$$:

Step 1: In this step, we take $$35$$ as a pair (by placing a bar over it). (If the number has an odd number of digits, then we place a bar just on the first digit; if the number has an even number of digits, then we place a bar on the first two digits together).

Step 2: Find a number whose square is very close to $$35$$ and less than or equal to $$35$$. We know that $$5^{2}=35$$. So $$5$$ is such a number. We write it in the place of both the quotient and the divisor.

Step 3: Since we do not have any other digits of $$35$$ to carry forward, we write pairs of zeros after the decimal point (as $$35 = 35.000000…$$). We write as many pairs as we want the number of decimals after the decimal point in the final result. Let us calculate $$\sqrt{35}$$ up to $$3$$ decimals. So, we write $$3$$ pairs of zeros. Since we have taken a decimal point in the dividend, let us write a decimal point in the quotient as well after $$5$$.

Step 4: Remember that we always carry forward two digits at a time while finding a square root. We carry forward two zeros at a time. Double the quotient and write it as the divisor of the next division. But, note that this is not the complete divisor.

Step 5: Now a part of the divisor is $$10$$, think which number should replace each of the boxes such that the product is very close to $$1000$$ and that is less than or equal to $$1000$$. We have $$109 \times 9 = 981$$. Thus, the required number is $$9$$. Include it in both the divisor and quotient.

Step 6: We repeat step 3 and step 4 for the corresponding divisors and quotients of the subsequent divisions.

Therefore, the square root of $$35$$ by long division method is $$5.916$$.

## Square Root of 35 by Babylonian Method or Hero’s Method

The Babylonian method (also known as Hero’s method) is an ancient method for finding square roots of numbers that is based on making an educated guess for the square root, and then repeatedly averaging that guess with the quotient of the original number divided by the current guess, in order to arrive at a better approximation.

Here is the steps for finding the square root of $$35$$ using the Babylonian method:

(Video) SQUARE ROOT OF 35

Step 1: Start with an initial guess for the square root, such as $$x_0 = 6$$.

Step 2: Divide the original number, $$35$$, by the current guess, $$6$$, to get the quotient: $$\frac{35}{6}=5.833…$$.

Step 3: Average the current guess and the quotient: $$\frac{(6 + 5.833…)}{2} = 5.917…$$.

Step 4: Use this new average as the new guess: $$x_1 = 5.917…$$.

Step 5: Repeat steps 2-4 until you achieve your desired level of accuracy.

Step 6: The square root of $$35$$ by Babylonian method is $$5.9161$$.

Notice that it is similar to the newton-raphson method, but the division is done by the number and the current guess. The approximation is similar to the one achieved by the long division method, which is $$5.916$$.

The Babylonian method is also efficient for finding square roots of a number that does not have an exact root, it can be used to find the square root of any positive number with a high degree of accuracy.

## Square Root of 35 by Newton Raphson Method

The Newton-Raphson method can also be used to find the square root of a number by treating the number as the solution to the equation $$x^2 = n$$. The Newton-Raphson method involves iteratively improving an initial guess for the solution, $$x_0$$, by using the formula:

$$x_{i+1}=x_i-\frac{f(x_i)}{f'(x_i)}$$, where $$f'(x)$$ is the derivative of $$f(x)$$ with respect to $$x$$.

To find the square root of $$35$$ using this method, you would use the equation $$f(x) = x^2 – 35$$ and repeatedly apply the formula above starting with an initial guess, $$x_0$$.

Here is an example of finding the square root of $$35$$ using the Newton-Raphson method:

Step 1: Take an initial guess, $$x_0 = 6$$.

Step 2: Calculate the $$x_1 = x_0 – f(x_0)/f'(x_0) = 6 – (6^2 -35)/(2\times 6) = 5.917$$.

Step 3: $$x_2 = x_1 – f(x_1)/f'(x_1) = 5.917 – (5.917^2 -35)/(2\times 5.917) = 5.9161$$.

Step 4: You can repeat steps 2 and 3 for desired accuracy.

(Video) Newton's Method

The solution will be the value of $$x$$ that makes $$f(x) = 0$$. The square root of $$35$$ by Newton-Raphson method is $$5.9161$$, which is similar to the results of previous methods.

Note that you can use any initial value as a starting point, the answer will still converge to the square root of $$35$$ but the number of iterations needed to reach the desired accuracy might change.

### Is Square Root of 35 Rational or Irrational?

The square root of $$35$$ is irrational. A number is considered rational if it can be expressed as the ratio of two integers, $$a$$ and $$b$$, where $$b$$ is not equal to zero. The decimal representation of a rational number either terminates (such as $$1/2 = 0.5$$) or repeats (such as $$1/3 = 0.3333…$$), while irrational numbers have a decimal representation that neither terminates nor repeats.

The square root of $$35$$ is a non-perfect square, and cannot be simplified to a whole number or ratio of two integers. Therefore, the decimal representation of its square root, which is approximately $$5.916079783099616$$, is non-repeating and non-terminating, thus it is considered an irrational number.

### Is Root 35 a Perfect Square?

No, $$35$$ is not a perfect square. A perfect square is a number that can be expressed as the square of an integer, such as $$4$$ ($$2^2$$) or $$9$$ ($$3^2$$).

To check if a number is a perfect square, one way is to find the square root of the number and check if the result is a whole number or not. In this case, the square root of $$35$$ is approximately $$5.916079783099616$$, which is not a whole number, it is an irrational number.

Another way to check if a number is a perfect square is to find the prime factorization of the number and check if all the exponents are even. A perfect square will have all the exponents to be even. For example, $$4 = 2^2$$ and $$36 = 2^2 \times 3^2$$ so they are perfect squares, but $$35 = 5\times 7$$, both exponents are not even so it is not a perfect square.

### Important Points on Square Root of 35

• The square root of $$35$$ is a quantity that when multiplied by itself will equal $$35$$.
• The square root of $$35$$ is $$\sqrt{35} = 5.916079$$.
• The square root of $$35$$ in exponential form is written as $$(35)^{\frac{1}{2}}$$.
• The square root of $$35$$ in radical form is written as $$\sqrt{35}$$.
• The square root of $$35$$ is an irrational number.
• $$35$$ is not a perfect square number.

## Solved Examples of Square Root of 35

Example 1. Find the solution of the equation $$x^2 – 35 = 0$$.

Solution: Given that:

$$x^2 – 35 = 0$$

$$x^2 = 35$$

$$x = \pm \sqrt {35}$$

$$x = \pm 5.916$$

Therefore, the solution for the given equation is $$\pm 5.916$$.

Example 2. A round pool has an area of $$35\pi$$ square feet. Find its radius. Round the answer to the nearest integer.

Solution: Let us assume that the radius of the pool is $$r$$ feet.

(Video) Approximating Square Roots w/ Newton's Method

We know that Area of a circle = $$\pi r^{2}$$.

Given that: $$\pi r^{2} = 35\pi$$

$$\Rightarrow$$ $$r^{2} = 35$$

By taking the square root on both sides, we get

$$\sqrt{r^{2}} = \sqrt{35}$$

$$\Rightarrow$$ $$r = 5.9$$

Thus, $$r = 5.9$$ and therefore, the radius of the pool is $$5.9$$ feet.

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

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## FAQs on Square Root of 35

Q.1What is the value of square root of 35?

Ans.1 An approximate value for the square root of 35 is 5.916079

Q.2Is the square root of 35 rational or irrational?

Ans.2 The square root of $$35$$ is an irrational number.

Q.3Is 35 a perfect square?

(Video) ApproxDerivEx2

Ans.3 $$35$$ is a perfect square if the square root of $$35$$ equals a whole number. As we know the square root of $$35$$ is not a whole number. So, $$35$$ is not a perfect square.

Q.4What is the square root of a number?

Ans.4 The square root of a number is a value that, when multiplied by itself, equals the given number. For example, the square root of $$9$$ is $$3$$, because $$3 \times 3 = 9$$.

Q.5What is the square root of 35 rounded to the nearest tenth?

Ans.5 To round the square root of $$35$$ to its nearest tenth means to have one digit after the decimal point. So, $$\sqrt{35} = 5.916 \approx 5.9$$.

(Video) 24.2 Newton's Method

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