Write each expression in terms of $i$ and simplify if possible.
a) $\sqrt{-100}=$ $\square$
b) $\sqrt{-49}=$ $\square$
c) $\sqrt{-252}=$ $\square$
Answer (2)
Express − 100 as 100 ⋅ − 1 and simplify to 10 i .
Express − 49 as 49 ⋅ − 1 and simplify to 7 i .
Express − 252 as 252 ⋅ − 1 , simplify 252 to 6 7 , and then write the expression as 6 7 i .
The final answers are 10 i , 7 i , and 6 7 i .
Explanation
Understanding the Problem We are asked to express the square roots of negative numbers in terms of i and simplify. Recall that i = − 1 .
Simplifying − 100 a) We have − 100 . We can rewrite this as 100 ⋅ − 1 = 100 ⋅ − 1 . Since 100 = 10 and − 1 = i , we have − 100 = 10 i .
Simplifying − 49 b) We have − 49 . We can rewrite this as 49 ⋅ − 1 = 49 ⋅ − 1 . Since 49 = 7 and − 1 = i , we have − 49 = 7 i .
Simplifying − 252 c) We have − 252 . We can rewrite this as 252 ⋅ − 1 = 252 ⋅ − 1 . To simplify 252 , we find the prime factorization of 252: 252 = 2 2 ⋅ 3 2 ⋅ 7 . Thus, 252 = 2 2 ⋅ 3 2 ⋅ 7 = 2 ⋅ 3 ⋅ 7 = 6 7 . Since − 1 = i , we have − 252 = 6 7 i .
Final Answer Therefore, the simplified expressions are: a) − 100 = 10 i b) − 49 = 7 i c) − 252 = 6 7 i
Examples
Complex numbers, which include imaginary numbers, are used in electrical engineering to analyze alternating current circuits. They help in representing and calculating impedance, which is the opposition to the flow of current in an AC circuit. Imaginary numbers are also used in quantum mechanics to describe the wave functions of particles. These functions are essential for predicting the behavior of subatomic particles.
The square roots of the negative numbers can be expressed in terms of i as follows: − 100 = 10 i , − 49 = 7 i , and − 252 = 6 7 i . This involves recognizing that the square root of a negative number can be represented by the square root of its absolute value multiplied by i . Each expression was simplified step-by-step using the properties of square roots and complex numbers.
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