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Question: How do you simplify the square root of negative \[6\] and root times square root of negative \[18\]?...

How do you simplify the square root of negative 66 and root times square root of negative 1818?

Explanation

Solution

In this question, we have to find out the required value from the given particulars.
We need to first find out the square root of negative 66 . Then we need to find out the square root of negative 1818 , then we will multiply the two values. After doing the multiplication we can find out the required solution.

Formula used: We know the formula for imaginary number,
i2=−1{i^2} = - 1
i.e., i=−1i = \sqrt { - 1}
Where i is the imaginary number.

Complete step-by-step solution:
We need to simplify the square root of negative 66 and root times square root of negative 1818 .
First, we need to find out the square root of negative 66 .
If we use, i=−1i = \sqrt { - 1} then we get,
−6=−1×6=−1.6=i6\sqrt { - 6} = \sqrt { - 1 \times 6} = \sqrt { - 1} .\sqrt 6 = i\sqrt 6 .
Now, we need to find out the square root of negative 1818 .
−18=−1×18=−1.18=i18\sqrt { - 18} = \sqrt { - 1 \times 18} = \sqrt { - 1} .\sqrt {18} = i\sqrt {18}
Here, the square root of negative 66 end root times square root of negative 1818
=−6×−18= \sqrt { - 6} \times \sqrt { - 18}
=i6×i18= i\sqrt 6 \times i\sqrt {18}
=i2618= {i^2}\sqrt 6 \sqrt {18}
=−108= - \sqrt {108} [Since for any real numbers, A.B=A.B\sqrt A .\sqrt B = \sqrt {A.B} ]
=−2×2×3×3×3= - \sqrt {2 \times 2 \times 3 \times 3 \times 3}
=−(±2×33)= - \left( { \pm 2 \times 3\sqrt 3 } \right)
=∓63= \mp 6\sqrt 3

Hence, simplifying the square root of negative 66 and root times square root of negative 1818 are either 636\sqrt 3 or, −63- 6\sqrt 3.

Note: For this problem, we need to know what it is i.
A complex number is a number that can be expressed in the form a+bi where a and b are real numbers and i represents the imaginary unit, satisfying the equation i2=−1{i^2} = - 1 . Since no real number satisfies this equation, i is called an imaginary number.
Square root:
In mathematics, a square root of a number x is a number y such that, y2=x{y^2} = x . In other words, a number y whose square is x.
For example, 4,−44, - 4 are square roots of 1616 , because 42=(−4)2=16{4^2} = {\left( { - 4} \right)^2} = 16 .