Square root of 45

Before we begin, let's understand the meaning of the square root. The symbol square root is written as √ and 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 square root of 45. We will do so using methods like prime factorization and division, and along with it, also find out the square root of 45.

• Square Root of 45: √45 = 6.708
• Square of 45: 45² = 2025

We know that addition has an inverse operation as subtraction and multiplication has an inverse operation as division. Similarly, finding the square root is an inverse operation of squaring. The square root of 45 is the number that gets multiplied to itself to give the number 45. So, we have to think of a number whose square is 45. We can see that, there does not exist any integer whose square is 45. The square root of a number is the number that gets multiplied to itself to give the original number. The square root of 45 is 6.7082039325.

It is not possible to break 45 into two such factors which, on squaring, give 45. It can be approximately written as a square of 6.708, which is a non-recurring and non-terminating decimal number. This shows that it is not a perfect square, which also proves that the square root of 45 is an irrational number. Do you think the square root of 45 and the square root of 5 have anything in common? Yes, there is, both are not perfect squares. So, √45 is an irrational number.

We can find the square root of 45 using the following steps:

• Step 1: Check whether the number is a perfect square or not. In this case, 45 is not a perfect square as it cannot be broken down into a product of two same numbers.
• Step 2: Once the number is checked, the following process needs to be followed:

1. If the number is a perfect square, it can be written in this format √x² = x
2. If it is not a perfect square, the square root is found using the long division method. It can also be written in its simplified radical form.

Since 45 is not a perfect square, we find its square root using the long division method. The simplified radical form of the square root of 45 is given below.

Simplified radical form of the square root of 45

45 can be written as a product 9 and 5. Hence, square root of 45 can be expressed as, √45 = √(9 × 5) =  3√5. 45 is not a perfect square, hence, it remains within roots. The simplified radical form of the square root of 45 is 3√5.

Square Root of 45 By Long Division

Let us understand the process of finding the square root of 45 by long division.

• Step 1: Pair the digits of the number starting from the right end, by placing a bar over them.
• Step 2: Now, we find a number such that its square gives a product that is less than or equal to this pair. The square of 6 is 36, which is less than 45. On subtracting 36 from 45, we get 9.
• Step 3: Now, we double the quotient and place it as the next divisor with a blank on its right. The double of 6 gives 12 and a blank is placed next to it. This blank needs to be filled by a number which will also become the next digit in the quotient. It should be filled with the largest possible digit, such that when the new divisor is multiplied with the new quotient, the product is less than or equal to the dividend.
• Step 4: Repeating the above steps, the quotient 67 is doubled, which gives 134, and a pair of 0s is added to the dividend.
• Step 5: The same steps are repeated and the digit which is written in the blank space is 8. The product of 13408 and 8 gives 1072. After subtracting 107264 from 110000, we get 2736. Finally, the square root of 45 is calculated to be 6.708 as shown:

Explore Square roots using illustrations and interactive examples

• Square Root of 64
• Square Root of 35
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Important Notes:

• 45 is not a perfect square, hence, its square root is an irrational number. This concludes that the square root of any number "n", which is not a perfect square, will always be an irrational number.

Think Tank:

• Can the value of a square root be negative as well? Hint: Think what is the square of a negative number.
• Is √-45 a real number? Hint: Think whether there is any real number whose square is negative.

What is the square root of 45?

The square root of 45 is 6.7082039325.

What is the square of 45?

The square of 45 is 2025.

How do you find the square root of 45?

We can find the square root of 45 by using the prime factorization and long division.

Is the square root of 45 an irrational number?

Yes, the square root of 45 is an irrational number.

What is the square root of 45 in simplified form?

The square root of 45 in simplified form is 3√5.