document.write( "Question 608069: Find the square roots of 4 times the square root of 3 +4i. Ex: k=0 and k=1 \n" ); document.write( "
Algebra.Com's Answer #383043 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I'll do it the algebra way, and then I'll do it the trigonometric way.\r\n" );
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document.write( "The ALGEBRA way:\r\n" );
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document.write( "4·sqrt(3+4i) = A + Bi,  where A and B are real.\r\n" );
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document.write( "16(3 + 4i) = A² + 2ABi + B²i²\r\n" );
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document.write( "48 + 64i = A² + B²i² + 2ABi\r\n" );
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document.write( "48 + 64i = A² + B²(-1) + 2ABi\r\n" );
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document.write( "48 + 64i = A² - B² + 2ABi\r\n" );
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document.write( "48 = A² - B²\r\n" );
document.write( "64 = 2AB\r\n" );
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document.write( "Solve the second equation for B:  B = \"64%2F%282A%29\" = \"32%2FA\"\r\n" );
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document.write( "Substitute in 48 = A² - B²\r\n" );
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document.write( "48 = A² - (\"32%2FA\")²\r\n" );
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document.write( "48 = A² - \"1024%2FA%5E2\")\r\n" );
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document.write( "48A² = A4 - 1024\r\n" );
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document.write( "0 = A4 - 48A - 1024\r\n" );
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document.write( "0 = (A² - 64)(A² + 16}}}\r\n" );
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document.write( "A² - 64 = 0;   A² + 16 = 0\r\n" );
document.write( "     A² = 64        A² = -16\r\n" );
document.write( "      A = ±8         A = ±4i (discard since A is real)\r\n" );
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document.write( "Using A = -8, = B = \"32%2FA\" = \"32%2F%28-8%29\" = -4\r\n" );
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document.write( "So one solution is A + Bi = -8 - 4i\r\n" );
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document.write( "Using A = 8, = B = \"32%2FA\" = \"32%2F8\" = 4\r\n" );
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document.write( "So the other solution is A + Bi = 8 + 4i\r\n" );
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document.write( "-------------------\r\n" );
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document.write( "The trigonometric way: \r\n" );
document.write( "   ______\r\n" );
document.write( "±4√3 + 4i\r\n" );
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document.write( "We'll just get  ±√3 + 4i and multiply by 4 when we finish:\r\n" );
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document.write( "Get 3 + 4i in trigonometric form:\r\n" );
document.write( "     _______    ______    __\r\n" );
document.write( "r = V3² + 4² = V9 + 16 = V25 = 5\r\n" );
document.write( "θ = arctan(\"4%2F3\" = 53.13° + k·360°\r\n" );
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document.write( "Trig form is r(cos(θ+k*\"350°\") + isin(θ+k*\"360°\")), \r\n" );
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document.write( "3 + 4i = 5(cos(53.13°+k·360°) + i·sin(53.13°+k·360))\r\n" );
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document.write( "\"+matrix%282%2C1%2C%22%22%2C%283%2B4i%29%5E%281%2F2%29%29\" = \r\n" );
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document.write( "\"+matrix%282%2C1%2C%22%22%2C%283%2B4i%29%5E%281%2F2%29%29\" = \"+matrix%282%2C1%2C%22%22%2C5%5E%281%2F2%29%29\" =\r\n" );
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document.write( " =\r\n" );
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document.write( ", where k = 0, 1.\r\n" );
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document.write( "Answer is 4 times that or:\r\n" );
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document.write( ", where k = 0, 1.\r\n" );
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document.write( "That works out to be:\r\n" );
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document.write( "When k = 0\r\n" );
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document.write( "\"4sqrt%285%29%28cos%28%2226.565%B0%22%29%2Bi%2Asin%28%2226.565%B0%22%29%29%29\", \r\n" );
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document.write( "4(2.000000893 + .999992136i) or\r\n" );
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document.write( "8.000003573+3.999992854i\r\n" );
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document.write( "which is very close to 8+4i.  The difference is because we\r\n" );
document.write( "rounded off the angle.\r\n" );
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document.write( "When k = 1\r\n" );
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document.write( "\"4sqrt%285%29%28cos%28%22206.565%B0%22%29%2Bi%2Asin%28%22206.565%B0%22%29%29%29\", \r\n" );
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document.write( "4(-2.000000893 - .999992136i) or\r\n" );
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document.write( "-8.000003573-3.999992854i\r\n" );
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document.write( "which is very close to -8-4i.\r\n" );
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document.write( "Edwin
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