How Do u Simplify The Square Root Of 175?

In mathematical terms, square root of any number, say ‘a’ can be denoted by the number y so as to say y² = a. This just means that, for any number y, the square is the result of multiplication of the number with the same that is y * y. Similarly, to get the square root, if a = y², then the square root of a will be y. As an example, either 4 or −4 can be the square roots of 16. In other words, squares of both 4 and -4 forms the number 16.

How to Square A Number

A number can be squared by multiplying the number with itself. For example: 3 squared can be written as:

3 Squared = 3 x 3 = 9 

"Squared" is often written as a little 2 like 4² which just means 4 squared equals 16. Here the little 2 signifies that the number appears two times in multiplication.

Negative Numbers

Negative numbers can also be squared. As an example, when we ‘−5’ gets squared, the result becomes:

−5 × −5 = 25 

This is because negative times of a negative integers always returns positive integers, very similar to squaring positive numbers.

Square Roots

Square roots of a number are just the opposite processes such that if 3 squared gives 9 then square root of 9 gives 3. Simply put square root of a number is the value which if multiplied by itself returns the original number. The square root of 9 is 3 since 3 when multiplied by itself gives 9. 

Here are some more squares and square roots:

The Square Root Symbol

The square root symbol ‘√’ is of the form of a tick and use of this symbol started several hundred years ago through a dot and an upward flick. The symbol is known as the radical and gives meaning to mathematical equations without excess explanations.

The square root symbol is used in the following manner:

Square root of 25 à √25 = 5

25 = 5 × 5 or the multiplication of 5 by itself and hence √25 gives 5

However, there is a catch the square root could have also been −5 too since −5 × −5 also equal 25. So, the answer is √25 can be both +5 or -5 but generally when the square root symbol is used the positive value is referred.

Perfect Squares

Perfect squares are referred to those "Square Numbers" (square of a number)  which are the squares of integers. As examples Perfect squares can be 4 of integer 2, 9 of integer 3, 16 of integer 4, 25 of integer 5, 36 of integer 6, 49 of integer 7, 64 of integer 8, 81 of integer 9, 100 of integer 10 and so on. Any number greater between these perfect squares that is greater than 2 except the perfect squares are not square of integers, which means they are squares of decimals or fractions.

Calculating Square Roots

It is always easier to figure out the square root of perfect squares, however it becomes painstaking to calculate other square roots.

An example is √10?

Now, 3 * 3 equals 9 and 4 * 4 equals 16, therefore the answer is surely between 3 and 4.

• If tried with 3.5 - 3.5 * 3.5 = 12.25
• If tried with 3.2 - 3.2 × 3.2 = 10.24
• If tried with 3.1 - 3.1 × 3.1 = 9.61

Thus, square root of 10 is between 3.1 and 3.2, but to get closer to a precise answer it will take large number of steps and a long time. If consulted with a calculator, it gives the result:

3.1622776601683793319988935444327

Yet the digits continue endlessly after the decimal point and there is no pattern to it. Hence, it should be noted that even this answer is a mere approximation only. 

Simplifying process of calculating Square Root

To simplify the calculation process and to make square roots get more accurate with every round, the following steps are devised.

a) To start with guess numbers (suppose 4 to be square root of 10)

b) Next to divide the square with the guess number (10/4 gives 2.5)

c) Now add this to guess number (4 + 2.5 gives 6.5)

d) Again, divide this with 2 or halve it (6.5/2 gives 3.25)

e) Next, this value is to be set as the new guess number, and start from b) again

• The first attempt from 4 gave 3.25
• Starting again (b to e) gave 3.163
• Starting again (b to e) gave 3.1623 

Now, after 3 times the answer 3.1623 is quite good since 3.1623² gives us

3.1623 * 3.1623 = 10.00014

To Find √175

Now the number 175 lies between two perfect squares, 196 and 169 of integers 14 and 13 respectively.

f) To start with guess numbers (suppose 14 to be square root of 175)

g) Next to divide the square with the guess number (175/14 gives 12.5)

h) Now add this to guess number (14 + 12.5 gives 26.5)

i) Again, divide this with 2 or halve it (26.5/2 gives 13.25)

j) Next, this value is to be set as the new guess number, and start from b) again

• The first attempt from 4 gave 13.25
• Starting again (b to e) gave 13.23
• Starting again (b to e) gave 13.229 

Now, after 3 times the answer 13.229 is quite good since 13.229² gives us

13.229 * 13.229 = 175.006441

Thus, the square root process has been studied here at great depth and the square root process to figure out the square root values of numbers that are not perfect squares have been simplified. The simplified process is first demonstrated with the number 10 and then applied on the number 175. It is observed that just three rounds are enough to provide near accurate results. This makes the simplified process far more effective and efficient than the brute force methods used by calculators.