Which expression is equivalent to $\sqrt{-108}-\sqrt{-3}$ ?
A. $51 \sqrt{3}$
B. $6i \sqrt{3}$
C. $7i \sqrt{3}$
D. $8i \sqrt{3}$
Answer (1)
Simplify − 108 to 6 i 3 .
Simplify − 3 to i 3 .
Subtract the simplified terms: 6 i 3 − i 3 = 5 i 3 .
The equivalent expression is 5 i 3 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to − 108 − − 3 . Let's simplify each term separately. Recall that the imaginary unit i is defined as i = − 1 .
Simplifying the First Term First, let's simplify − 108 . We can rewrite this as 108 ⋅ − 1 . Since 108 = 36 ⋅ 3 , we have 108 = 36 ⋅ 3 = 36 ⋅ 3 = 6 3 . Therefore, − 108 = 6 3 i .
Simplifying the Second Term Next, let's simplify − 3 . We can rewrite this as 3 ⋅ − 1 = 3 i .
Substituting Back into the Original Expression Now, substitute these simplified expressions back into the original expression: 6 3 i − 3 i .
Combining Like Terms Combine like terms: ( 6 3 − 3 ) i = 5 3 i .
Final Answer Therefore, the expression equivalent to − 108 − − 3 is 5 3 i or 5 i 3 .
Examples
Complex numbers, which involve the imaginary unit i = − 1 , are used in electrical engineering to analyze alternating current (AC) circuits. Impedance, which is the opposition to the flow of current in an AC circuit, is represented as a complex number. Simplifying expressions with square roots of negative numbers, as we did in this problem, is a fundamental skill needed to work with impedances and analyze AC circuits.
