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Question 1 of 3
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Imaginary numbers have the properties i=√-1i=√−1 or i2=-1i2=−1.
To solve 13-i-23+i13−i−23+i, first find a common denominator for the two fractions.
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13-i-23+i13−i−23+i |
Multiply the numerator and the denominator of the first fraction by the denominator of the second fraction 3+i3+i. |
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13-i×3+i3+i-23+i13−i×3+i3+i−23+i |
Simplify the numerator only. |
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3+i(3-i)(3+i)-23+i×3-i3-i3+i(3−i)(3+i)−23+i×3−i3−i |
Multiply the numerator and the denominator of the second fraction by the original denominator of the first fraction 3-i3−i. |
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3+i(3-i)(3+i)-6-2i(3-i)(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. |
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3-6+i+2i(3-i)(3+i)3−6+i+2i(3−i)(3+i) |
Simplify the numerator. |
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-3+3i(3-i)(3+i)−3+3i(3−i)(3+i) |
Simplify the denominator by remembering that a2-b2=(a+b)(a-b)a2−b2=(a+b)(a−b) where a=3a=3 and b=ib=i. |
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-3+3i(3)2-(i)2−3+3i(3)2−(i)2 |
Simplify |
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-3+3i9-i2−3+3i9−i2 |
Remember that i2=-1i2=−1 |
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-3+3i9-(-1)−3+3i9−(−1) |
Simplify |
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-3+3i10−3+3i10 |
Divide both terms in the numerator by 1010. |
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-310+310i−310+310i |
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Question 2 of 3
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Imaginary numbers have the properties i=√-1i=√−1 or i2=-1i2=−1.
To solve 8i38i3, multiply the numerator and the denominator by ii and then simplify.
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8i38i3 |
Multiply the numerator and the denominator by ii. |
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8i3×ii8i3×ii |
Simplify |
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8ii48ii4 |
Remember that i2×i2=i4i2×i2=i4. |
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8ii2×i28ii2×i2 |
Remember that i2=-1i2=−1. |
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8i(-1)×(-1)8i(−1)×(−1) |
Simplify |
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8i18i1 |
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8i8i |
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Question 3 of 3
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Imaginary numbers have the properties i=√-1 or i2=-1.
To simplify 72+i√2, multiply the numerator and the denominator by the conjugate of the denominator 2-i√2.
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72+i√2 |
Multiply the numerator and the denominator by the conjugate of the denominator 2-i√2. |
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72+i√2×2-i√22-i√2 |
Simplify the numerator only. |
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14-7i√2(2+i√2)(2-i√2) |
Simplify the denominator by remembering that a2-b2=(a+b)(a-b) where a=2 and b=i√2. |
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14-7i√2(2)2-(i√2)2 |
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14-7i√24-2i2 |
Remember that i2=-1 |
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14-7i√24-2(-1) |
Simplify the denominator. |
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14-7i√24+2 |
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14-7i√26 |
Divide both terms in the numerator by 6. |
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146-7i√26 |
Simplify the first term. |
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73-7i√26 |
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