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Mixed Operations with Complex NumbersMixed Operations with Complex Numbers
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Question 1 of 3
1. Question
Solve
`(1)/(3-i)-(2)/(3+i)`
Correct
Great Work!
Incorrect
Imaginary numbers have the properties `i=sqrt(-1)` or `i^2=-1`.To solve `(1)/(3-i)-(2)/(3+i)`, first find a common denominator for the two fractions.`(1)/(3-i)-(2)/(3+i)` Multiply the numerator and the denominator of the first fraction by the denominator of the second fraction `3+i`. `=` `(1)/(3-i)timescolor(red)(3+i)/color(red)(3+i)-(2)/(3+i)` Simplify the numerator only. `=` `(3+i)/((3-i)(3+i))-(2)/(3+i)timescolor(blue)(3-i)/color(blue)(3-i)` Multiply the numerator and the denominator of the second fraction by the original denominator of the first fraction `3-i`. `=` `(3+i)/((3-i)(3+i))-(6-2i)/((3-i)(3+i))` Subtract the numerators and place this answer over the common denominator. `=` `(3-6+i+2i)/((3-i)(3+i))` Simplify the numerator. `=` `(-3+3i)/((3-i)(3+i))` Simplify the denominator by remembering that `a^2-b^2=(a+b)(a-b)` where `a=3` and `b=i`. `=` `(-3+3i)/((3)^2-(i)^2)` Simplify `=` `(-3+3i)/(9-i^2)` Remember that `i^2=-1` `=` `(-3+3i)/(9-(-1))` Simplify `=` `(-3+3i)/(10)` Divide both terms in the numerator by `10`. `=` `-3/10+3/10i` `-3/10+3/10i` -
Question 2 of 3
2. Question
Solve
`(8)/(i^3)`
Correct
Great Work!
Incorrect
Imaginary numbers have the properties `i=sqrt(-1)` or `i^2=-1`.To solve `(8)/(i^3)`, multiply the numerator and the denominator by `i` and then simplify.`(8)/(i^3)` Multiply the numerator and the denominator by `i`. `=` `(8)/(i^3)timescolor(red)(i)/color(red)(i)` Simplify `=` `(8i)/(i^4)` Remember that `i^2timesi^2=i^4`. `=` `(8i)/(i^2timesi^2)` Remember that `i^2=-1`. `=` `(8i)/((-1)times(-1))` Simplify `=` `(8i)/(1)` `=` `8i` `8i` -
Question 3 of 3
3. Question
Simplify
`(7)/(2+isqrt(2))`
Correct
Great Work!
Incorrect
Imaginary numbers have the properties `i=sqrt(-1)` or `i^2=-1`.To simplify `(7)/(2+isqrt(2))`, multiply the numerator and the denominator by the conjugate of the denominator `2-isqrt(2)`.`(7)/(2+isqrt(2))` Multiply the numerator and the denominator by the conjugate of the denominator `2-isqrt(2)`. `=` `(7)/(2+isqrt(2))timescolor(red)(2-isqrt(2))/color(red)(2-isqrt(2))` Simplify the numerator only. `=` `(14-7isqrt(2))/((2+isqrt(2))(2-isqrt(2)))` Simplify the denominator by remembering that `a^2-b^2=(a+b)(a-b)` where `a=2` and `b=isqrt(2)`. `=` `(14-7isqrt(2))/((2)^2-(isqrt(2))^2)` `=` `(14-7isqrt(2))/(4-2i^2)` Remember that `i^2=-1` `=` `(14-7isqrt(2))/(4-2(-1))` Simplify the denominator. `=` `(14-7isqrt(2))/(4+2)` `=` `(14-7isqrt(2))/(6)` Divide both terms in the numerator by `6`. `=` `14/6-(7isqrt(2))/(6)` Simplify the first term. `=` `7/3-(7isqrt(2))/(6)` `7/3-(7isqrt(2))/(6)`
Quizzes
- Simplify Roots of Negative Numbers 1
- Simplify Roots of Negative Numbers 2
- Powers of the Imaginary Unit 1
- Powers of the Imaginary Unit 2
- Solve Quadratic Equations with Complex Solutions 1
- Solve Quadratic Equations with Complex Solutions 2
- Equality of Complex Numbers
- Add and Subtract Complex Numbers 1
- Add and Subtract Complex Numbers 2
- Multiply Complex Numbers 1
- Multiply Complex Numbers 2
- Divide Complex Numbers
- Complex Numbers – Product of Linear Factors 1
- Complex Numbers – Product of Linear Factors 2
- Mixed Operations with Complex Numbers