Simplify 4/(9-i)

Math
49-i
Multiply the numerator and denominator of 49-i by the conjugate of 9-i to make the denominator real.
49-i⋅9+i9+i
Multiply.
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Combine.
4(9+i)(9-i)(9+i)
Simplify the numerator.
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Apply the distributive property.
4⋅9+4i(9-i)(9+i)
Multiply 4 by 9.
36+4i(9-i)(9+i)
36+4i(9-i)(9+i)
Simplify the denominator.
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Expand (9-i)(9+i) using the FOIL Method.
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Apply the distributive property.
36+4i9(9+i)-i(9+i)
Apply the distributive property.
36+4i9⋅9+9i-i(9+i)
Apply the distributive property.
36+4i9⋅9+9i-i⋅9-ii
36+4i9⋅9+9i-i⋅9-ii
Simplify.
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Multiply 9 by 9.
36+4i81+9i-i⋅9-ii
Multiply 9 by -1.
36+4i81+9i-9i-ii
Raise i to the power of 1.
36+4i81+9i-9i-(i1i)
Raise i to the power of 1.
36+4i81+9i-9i-(i1i1)
Use the power rule aman=am+n to combine exponents.
36+4i81+9i-9i-i1+1
Add 1 and 1.
36+4i81+9i-9i-i2
Subtract 9i from 9i.
36+4i81+0-i2
Add 81 and 0.
36+4i81-i2
36+4i81-i2
Simplify each term.
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Rewrite i2 as -1.
36+4i81–1
Multiply -1 by -1.
36+4i81+1
36+4i81+1
Add 81 and 1.
36+4i82
36+4i82
36+4i82
Cancel the common factor of 36+4i and 82.
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Factor 2 out of 36.
2(18)+4i82
Factor 2 out of 4i.
2(18)+2(2i)82
Factor 2 out of 2(18)+2(2i).
2(18+2i)82
Cancel the common factors.
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Factor 2 out of 82.
2(18+2i)2⋅41
Cancel the common factor.
2(18+2i)2⋅41
Rewrite the expression.
18+2i41
18+2i41
18+2i41
Split the fraction 18+2i41 into two fractions.
1841+2i41
Simplify 4/(9-i)

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