Question

How do you find the principal square root of \[8\]?

Answer 1

Hint:
For the square root of given number, firstly, we will simplify the expression of the square root of given number involves finding factors. This means we are trying to find the two whole numbers that when we multiply, we get the number. Then continue factoring until you have all prime numbers. To simplify a number underneath the square root symbol, it is very useful to keep factoring the factors until the only factors that are left are prime numbers. Combine factors using exponents. Then, move bases outside of square root if possible, then rule for moving bases outside of the symbol is to divide the exponent power \[2\].

Complete Step by step Solution:
Step 1: We find the square root of \[8\]. Square root of \[8\] can be written as \[\sqrt 8 \]. Firstly, we find the number when it multiplied, we get \[8\]. \[2\] and \[4\] are two numbers which when multiplied, we get \[8\]. So, we can write \[\sqrt 8 = \sqrt {2 \times 4} \].
Step 2: Further find the prime number (factors) because it is very useful to keep factoring the factors until the only factors that are left are prime numbers.
So, \[4\] can be written as \[4 = 2 \times 2\] and
\[2\] can be written as \[2 = 2 \times 1\].
Step 3: Then the repeated factors can be rewritten more efficiently by using exponents.
Step 4: Further, we can give the underneath root symbol to each of factors (numbers).
Therefore, \[\sqrt 8 = \sqrt {2 \times 2 \times 2} = \sqrt {{{\left( 2 \right)}^2} \times 2} \]
After solving it, we get: \[\sqrt 8 = \sqrt {{{\left( 2 \right)}^2}} \times \sqrt 2 \]
We can write it as: \[\sqrt 8 = 2\sqrt 2 \]
We know that the \[\sqrt 2 \] is approximately \[1.41\]
Therefore, \[\sqrt 8 = 2 \times 1.41 = 2.82\]
Hence, \[\sqrt 8 = 2.82\]

The principal square root of \[8\] is \[2.82\].

Note:
Square root is the inverse option of squaring. The positive square root of a number is denoted by the symbol \[\sqrt {} \]. Example: \[\sqrt 9 = 3\]. To find the square root of a decimal number, we put bars on the integral part of the number in the usual manner.