urgent math question please help!

Question

Super confused on these two problems I'd really appreaciate it if someone helped me to understand and complete them!

1. write sqrt16 * 2^1/4 as a single power of 2. carefully show each step of your work.
2. Complex numbers are used by engineers and physicists to measure electrical circuits. To measure voltage, they use the formula E = I • Z, where E is voltage in volts, I is the current in amps, and Z is impedance, or resistance, in ohms.

a. Find the voltage for an electrical circuit if the current is 3 - 4i amps and the impedance is 2 + i ohms. Simplify your answer completely.  

1. b. Find the Impedance for an electrical circuit if the voltage is 18 - 24i and the current is 6 - 3i amps. Simplify your answer completely.

Answer 1

1. write sqrt16 * 2^1/4 as a single power of 2. carefully show each step of your work.

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1. ANSWER: sqrt 16 = 4 = 2^2; then multiply this by 2^ 1/4: Express in power of 2. First, take 2^2 * 2^1/4. This is a problem like x^a * x^b = x^(a+b). Follow this format to get: 2^2 * 2^1/4 = 2^ (2+ 1/4), so final answer is 2^(2 1/4)/

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1. Complex numbers are used by engineers and physicists to measure electrical circuits. To measure voltage, they use the formula E = I • Z, where E is voltage in volts, I is the current in amps, and Z is impedance, or resistance, in ohms.

a. Find the voltage for an electrical circuit if the current is 3 - 4i amps and the impedance is 2 + i ohms. Simplify your answer completely.  

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ANSWER: E = I * Z. Use I = 3-4i and Z = 2+i. Thus, E = (3-4i)(2+i). Use "FOIL" to get E = (3*2) + (-4i*2) + (3*i) + (-4i*i). This is 6 + (-8i) + (3i) + (-4*(i^2))). Recall i^2 = -1. So this is 6 + (-8+3)i + (-4*-1). Then, you have E = 10 - 5i. Maybe they want it expressed as: 5(2-i)

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1. b. Find the Impedance for an electrical circuit if the voltage is 18 - 24i and the current is 6 - 3i amps. Simplify your answer completely.

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ANSWER: E = I * Z. Find Z. This is Z = E/I. Substitute: Z = 18-24i/6-3i. This is 6(3-4i) / 6-3i. Multiply both numerator and denominator by the conjugate, which is 6+3i. This yields: 6(3-4i) / (6-3i)*(6+3i). This is 6(3-4i) / 36 - 9i^2. The denominator is 36+9 = 45 = 3*15. Thus, you have 6(3-4i)/(3*15) = 2(3-4i)/15. This is written (6-8i)/15.

Answer 2

First recall some properties of exponents...

x^a * x^b = x^(a+b) <--- multiplies monomials with same base

you add the exponents

(x^b)^(a) = x^(b^a) <--- monomial to a power, you multiply the exponents

and

sqrt(x) = x^(1/2)

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#1)

Do you mean sqrt(16) * 2^(1/4) or sqrt[(16) * 2^(1/4)]

case #1: sqrt(16) * 2^(1/4)

= 4 * 2^(1/4)

= 2^2 * 2^(1/4)

= 2^( 2 and 1/4)

= 2^(9/4) <--- same as Robert

case #2:

sqrt[ 16 * 2^(1/4) ] =

sqrt[16] * sqrt(2^(1/4)) =

4 * sqrt(2^(1/4)) =

2^2 * 2^[1/4*1/2] =

2^2 * 2^(1/8) =

2^[2 and 1/8] =

2^[17/8]

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#2) E = I z

= (3 - 4i)(2+i)

= 6 + 3i - 8i - (-4))

= 10 - 5i

#3) E = I z ---> z = E/I

z = (18 - 24i) 6 + 3i

----------- * -----------

6 - 3i 6 + 3i

= 108 + 54i - 144i + 72

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36 + 9

= 180 - 90i

------------

45

= 4 - 2i

CHECK: (4-2i)(6 - 3i) = 24 - 12i - 12i - 6

= 18 - 24i

Answer 3

1.) You have:

sqrt(16) * 2^1/4 = 4 * 2^1/4 = 2² * 2^1/4 = 2^5/4

2a.) Simply plug in the values I = 3 - 4i and Z = 2 + i to get:

V = (3 - 4i)(2 + i) = 6 - 5i + 4 = 10 - 5i

2b.) Since E = IZ, solving for I yields I = E / Z, therefore we have:

[multiply numerator and denominator by conjugate of Z]

I = (18 - 24i) / (6 - 3i) = (6 - 8i) / (2 - i) = (6 - 8i)(2 + i) / [(2 - i)(2 + i)] = (12 + 10i + 8)/5 = (20 + 10i)/5

= 4 + 2i