Divide the following complex numbers. (8+19 i)/(4-3i)

Write Problem: Write down the problem to solve. Divide the complex numbers (8+19i) by (4-3i).Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of (4-3i) is (4+3i). We multiply both the numerator and the denominator by this conjugate to eliminate the imaginary part in the denominator. ((8+19i)(4+3i)) / ((4-3i)(4+3i))Expand Numerator: Expand the numerator using the distributive property (FOIL method). (8+19i)(4+3i) = 8*4 + 8*3i + 19i*4 + 19i*3i = 32 + 24i + 76i + 57i^2 Since i^2 = -1, replace 57i^2 with -57. = 32 + 24i + 76i - 57 = (32 - 57) + (24i + 76i) = -25 + 100iExpand Denominator: Expand the denominator using the difference of squares. (4-3i)(4+3i) = 4^2 - (3i)^2 = 16 - 9i^2 Since i^2 = -1, replace -9i^2 with 9. = 16 + 9 = 25Divide Simplified: Divide the simplified numerator by the simplified denominator. (-25 + 100i) / 25 = (-25/25) + (100i/25) = -1 + 4i

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Algebra 2

Add, subtract, multiply, and divide polynomials

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Q. Divide the following complex numbers.

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\frac{8+19 i}{4-3 i}

Write Problem: Write down the problem to solve. $\newline$ Divide the complex numbers $(8+19i)$ by $(4-3i)$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ The conjugate of $(4-3i)$ is $(4+3i)$ . We multiply both the numerator and the denominator by this conjugate to eliminate the imaginary part in the denominator. $\newline$ $\frac{(8+19i)(4+3i)}{(4-3i)(4+3i)}$
Expand Numerator: Expand the numerator using the distributive property (FOIL method). $\newline$ $(8+19i)(4+3i) = 8\cdot 4 + 8\cdot 3i + 19i\cdot 4 + 19i\cdot 3i$ $\newline$ $= 32 + 24i + 76i + 57i^2$ $\newline$ Since $i^2 = -1$ , replace $57i^2$ with $-57$ . $\newline$ $= 32 + 24i + 76i - 57$ $\newline$ $= (32 - 57) + (24i + 76i)$ $\newline$ $= -25 + 100i$
Expand Denominator: Expand the denominator using the difference of squares. $\newline$ $(4-3i)(4+3i) = 4^2 - (3i)^2$ $\newline$ $= 16 - 9i^2$ $\newline$ Since $i^2 = -1$ , replace $-9i^2$ with $9$ . $\newline$ $= 16 + 9$ $\newline$ $= 25$
Divide Simplified: Divide the simplified numerator by the simplified denominator. $\newline$ $(-25 + 100i) / 25$ $\newline$ = $(-25/25) + (100i/25)$ $\newline$ = $-1 + 4i$