Before we begin, let's understand the meaning of square root. The symbol of square root is√and it is an integral part of mathematics. Once you understand the basics of finding the square root of a number, you can solve any square root-related problem. In this short lesson, we will learn about the squareroot of 65 and learn how to find it using different methods like the long division method.
Let us find the square root of 65.
The square root of a number is the number which, when multiplied by itself, givesthe original number as the product. The square root of 65 is 8.0622577483. The square root of 65 in the radical form is expressed as√65 and in theexponent form, it is expressed as 651/2. The square root of 65rounded to 5 decimal places is8.06225.
Is the Square Root of 65Rational orIrrational?
Any number which cannot be expressed as p/q,where p and q are integers and q is not equal to 0, are calledirrational numbers. Can√65 be expressed in such a way? In the decimal form, we can see that the decimal part of √65is8.0622577482986...which is non-terminating,non-repeating, andnever-ending. Hence,√65 is an irrational number.
How to Find the Square Root of 65?
65 is not a perfect square. We can find the square root of 65 by the approximation method.To find theaccurate value, we can use the long division method. In the approximation method, we find square numbers close to 65. We see that 64 and 81 are the perfect square numbers close to 65. The square root of 64 is 8 and the square root of 81 is 9. Therefore, the square root of 65 must lie between 8 and 9 and it must be closer to 8 as 65 is closer to 64. This method only gives us an approximate answer. To know the exact value, we can use the long division method and find a more accurate decimal value for√65.
Simplified Radical Form of Square Root of 65
√65in the simplest radical form is√65only. We cannot express √65as √(a×b), where a and b areintegers.Thus,√65 is already in the simplest radical form.
Square Root of 65ByLong Division
Let us follow these steps to find the square root of 65by long division.
- Step 1:Set up the number in pairs of two digits by placing a bar above it, starting from the unit's place.
- Step 2:Find a divisor such that when we multiply it to itself, the product is <=65. We see that 8 ×8 = 64
- Step 3:Since we do not have any other digits after 65to carry forward, we write pairs of zeros after the decimal point (as 65 = 65.000000...). Since we have added a decimal point in the dividend, let us include a decimal point in the quotient after 8.
- Step 4:Bring the 0 pair downand our new dividend is 100. We need to find a new divisor. We double the quotient8 × 2 = 16 and find a new digit for our unitsplacefor the new divisor, such that the product of this new divisor with that number is less than or equal to 100. Here we will place0 asthe units digit for our new divisor and also place it after the decimal point in the quotient. Our new divisor is 160.
- Step 5:We bring down the next pair of zeros and our new dividend is 10000. We need to find a new divisor like how we did in the previous step. We double the quotient8 × 2 = 16 and find the units place of our new divisor such that the productis less than or equal to 10000. We see that 1606 × 6=9636 which is close to 10000. Thus, we place 6 at the units place of our new divisor and place 6 in the quotient.
Bring the next pair of zeros down. We will repeat the process to find the next decimal place. The square root of 65 up to 2 decimal placesis 8.06as shown above. Can you continue the process and find the square root of 65 to 5 decimal places?
Explore square roots using illustrations and interactive examples
- Square Root of 63
- Square Root of 36
- Square Root of 28
- Square Root of 25
- Square Root of 68
Important Notes:
- The square root of 65in the radical form is expressed as√65.
- In the exponent form, the square root of 65is expressed as 651/2.
- The real roots of √65 are8.0622577.
Challenging Questions:
- What is the value of√√65?
- Determine the square root of 655bythe long division method.
Square Root of 65Solved Examples
Example 1:Jessica told herfriends thatthe value of-√65is the same as √-65. What can you say about her statement?
Solution
The square root of a negative number is an imaginary number. However,-√65 is a real number. Hence,they are not the same.
Example 2:Sam wants to paint the ceilingof his bedroom. The area of the ceiling is 65 sq. feet. If the roofis in the shape ofa square, what is the length of the side of theceiling?
Solution
Thearea of the ceiling is 65 sq. feet
Since, area = side×side
65 = 8.06× 8.06
√65= 8.06
Hence,the length of theside of the ceiling will be 8.06feet.
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FAQs on Square Root of 65
What is the square root of 65?
The square root of 65is8.0622577.
What is the square of 65?
The square of 65is 4225.
How do you find the square root of 65?
We can find the square root of 65by using the prime factorization method, repeated subtraction, or the long division method.
Is the square root of 65an irrationalnumber?
Yes, the square root of 65is an irrational number.
What is the square root of 65in simplified form?
The square root of 65in simplified form is√65.
