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Question 1 of 3
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Imaginary numbers have the properties i=√−1 or i2=−1.
To solve 13−i−23+i, first find a common denominator for the two fractions.
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13−i−23+i |
Multiply the numerator and the denominator of the first fraction by the denominator of the second fraction 3+i. |
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13−i×3+i3+i−23+i |
Simplify the numerator only. |
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= |
3+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−i. |
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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) |
Simplify the numerator. |
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= |
−3+3i(3−i)(3+i) |
Simplify the denominator by remembering that a2−b2=(a+b)(a−b) where a=3 and b=i. |
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= |
−3+3i(3)2−(i)2 |
Simplify |
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−3+3i9−i2 |
Remember that i2=−1 |
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= |
−3+3i9−(−1) |
Simplify |
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= |
−3+3i10 |
Divide both terms in the numerator by 10. |
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−310+310i |
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Question 2 of 3
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Imaginary numbers have the properties i=√−1 or i2=−1.
To solve 8i3, multiply the numerator and the denominator by i and then simplify.
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8i3 |
Multiply the numerator and the denominator by i. |
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8i3×ii |
Simplify |
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8ii4 |
Remember that i2×i2=i4. |
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8ii2×i2 |
Remember that i2=−1. |
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8i(−1)×(−1) |
Simplify |
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8i1 |
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8i |
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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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