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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Express the given complex numbers in polar and rectangular forms.3.00e0.500j
(a) Simplify (x2 − 4x + 4)1/2 . (b) For what values of x is your answer in part (a) valid? Explain.
Evaluate (819)12 /(816)14. What happens when you try to evaluate this on a calculator?
Express in terms of j: (a) −√−64 (b) −j15.
Perform the indicated operations, expressing all answers in the form a + bj.(3 − 7j) + (2 − j)
Express the given numbers in exponential form.0.450(cos 282.3° + j sin 282.3°)
Express each number in terms of j.√−81
Represent each complex number graphically and give the polar form of each.30 − 40j
Determine each of the following as being either true or false. If it is false, explain why.(2∠120°)3 = 8
Locate the given numbers in the complex plane.−4 − 3j
Perform the indicated operations. Leave the result in polar form.[4(cos60° + j sin60°)][2(cos20° + j sin 20°)]
Add graphically: (4 − 3j) + (−1 + 4j).
Represent each complex number graphically and give the polar form of each.−2.00 + 3.00j
Express 2.56(cos125.2° + j sin125.2°) in exponential form.
Perform the indicated operations. Leave the result in polar form.(0.5∠140°)(6∠110°)
An ac circuit contains the given combination of circuit elements from among a resistor (R = 45.0Ω), a capacitor (C = 86.2 μF), and an inductor (L = 42.9 mH). If the frequency in the circuit is f =
Express each number in terms of j.√−49
Perform the indicated operations, expressing all answers in the form a + bj.(5.4 − 3.4j) − (2.9j + 5.5)
Express the given numbers in exponential form.1672[cos(−7.14°) + j sin(−7.14°)]
Represent each complex number graphically and give the polar form of each.7.00 − 5.00j
Perform the indicated operations, expressing all answers in simplest rectangular form.(12 + 7j) + (−8 + 6j)
Locate the given numbers in the complex plane.3 − 4j
For an ac circuit in which R = 3.50Ω, XL = 6.20Ω, and XC = 7.35Ω, find the impedance and the phase angle between the current and the voltage.
Perform the indicated operations. Leave the result in polar form.(0.4∠320°)(5.5∠−150°)
An ac circuit contains the given combination of circuit elements from among a resistor (R = 45.0Ω), a capacitor (C = 86.2 μF), and an inductor (L = 42.9 mH). If the frequency in the circuit is f =
Express each number in terms of j.√−0.36
Perform the indicated operations, expressing all answers in the form a + bj.0.23 − (0.46 − 0.19j) + 0.67j
Express the given numbers in exponential form.0.515∠198.3°
Represent each complex number graphically and give the polar form of each.−0.55 − 0.24j
Perform the indicated operations, expressing all answers in simplest rectangular form.(18 − 3j) − (12 − 5j)
Perform the indicated operations graphically. Check them algebraically.2 + (3 + 4j)
Perform the indicated operations, expressing all answers in the form a + bj.(3 − 7j)2
Find the values of x and y that satisfy the given equations.2x −6xj3−3j2 = yj − y + 7j5
Solve the given problems. Refer to Example 4.In an alternating-current circuit, two impedances Z1 and Z2 have a total impedance ZT of Find ZT for Z1 = 3.2 + 4.8j mΩ and Z2 = 4.8 −
Solve the given problems. Refer to Example 4.If E = 5.70 − 3.65 j mV and I = 0.360 − 0.525 j μA, find the complex-number representation for Z.Data from Example 4In an alternating-current
Solve the given problems. Refer to Example 4.If E = 85 + 74j volts and Z = 2500 − 1200j ohms, find the complex-number representation for I.Data from Example 4In an alternating-current circuit, the
Solve the given problems. Refer to Example 4.If I = 0.835 − 0.427j amperes and Z = 250 + 170j ohms, find the complex-number representation for E.Data from Example 4In an alternating-current
Perform the indicated operations. Leave the result in polar form. 4/206° 100-320°
Perform the indicated operations. Leave the result in polar form.7644∠294.36° − 6871∠17.86°
Find the values of x and y that satisfy the given equations.x −2j2 + 7j = yj = 2xj3
Perform the indicated operations. Leave the result in polar form.17.8∠110.4° − 14.9∠226.3°
Perform the indicated operations. Leave the result in polar form. 245.6/326.44° 17.19/192.83°
Find the values of x and y that satisfy the given equations.9 − j = xj + 1 − y
Perform the indicated operations. Leave the result in polar form.0.983∠47.2° + 0.366∠95.1°
Find the values of x and y that satisfy the given equations.6j − 7 = 3 − x − yj
Perform the indicated operations. Leave the result in polar form. - 18(cos403° + jsin 403°) [2(cos 96° + jsin 96°)]²
In a microprocessor circuit, the current is I = 3.75∠15.0° μA and the impedance is Z = 2500∠−35.0° ohms. Find the voltage E in rectangular form. Use E = IZ.
Find the values of x and y that satisfy the given equations.2x + 3jy = −6 + 12j
When finding the current in a certain electric circuit, the expression (s + 1 + 4j)(s + 1 − 4j) occurs. Simplify this expression.
The voltage across a certain inductor is V = (8.66∠90.0°)(50.0∠135.0°) (10.0∠60.0°) volts. Simplify this expression and find the magnitude of the voltage.
Find the values of x and y that satisfy the given equations.7x − 2yj = 14 + 4j
Perform the indicated operations. Leave the result in polar form. 24(cos165° + jsin 165°) [3(cos 55° + jsin 55°)]³
If f(x) = x + 1/x, find f(1 + 3j).
The displacement d (in in.) of a weight suspended on a system of two springs is d = 6.03∠22.5° + 3.26∠76.0° in. Perform the addition and express the answer in polar form.
Find the conjugate of each complex number.(a) 6 (b) −5j
For 3/5 + 4/5 j, find: (a) The conjugate; (b) The reciprocal.
Solve for x: (x + 3 j)2 = 7 − 24j
The electric power p (in W) supplied to an element in a circuit is the product of the voltage e and the current i (in A). Find the expression for the power supplied if e = 6.80∠56.3° volts and i =
Find the conjugate of each complex number.(a). 2j (b). −4
Solve the given problems. Write j−2 + j−3 in rectangular form.
The cube roots of −1 can be found by solving the equation x3 + 1 = 0. Find these roots by factoring x3 + 1 as the sum of cubes and compare with Example 5.Data from Example 5Find the cube root of
Solve for x: (x + 2j)2 = 5 + 12j
Perform the indicated operations. Leave the result in polar form.(0.1254∠172.38°)(27.17∠204.34°)
Perform the indicated operations and simplify each complex number to its rectangular form. 129√-4 4
The cube roots of 8 can be found by solving the equation x3 − 8 = 0. Find these roots by factoring x3 − 8 as the difference of cubes and compare with Exercise 42.Data From Exercises 42In
In Exercises find the conjugate of each complex number.(a) 2j − 3 (b) −9 − j
In Example 5, we showed that one cube root of −1 is 1/2 - 1/2j√3 Cube this number in rectangular form and show that the result is −1.Data from Example 5Find the cube root of −1. Because −1
Perform the indicated operations. Leave the result in polar form.(40∠18°)(0.5∠245°)
Find the conjugate of each complex number.(a) 6 − 7j(b) 8 + j
Solve the given problems. Write the reciprocal of 2 + 5j in rectangular form.
Using the results of Example 6, find the square roots of 32j.Data from Example 6Find the two square roots of 2j.First, we write 2j in polar form as 2j = 2(cos 90° + j sin 90°). To find square
Perform the indicated operations. Leave the result in polar form.[2.5(cos162° + j sin162°)][8(cos115° + j sin115°)]
Perform the indicated operations and simplify each complex number to its rectangular form. 10 - √75 10
Perform the indicated operations and simplify each complex number to its rectangular form. √-9-6 3
Explain why the two square roots of a complex number are negatives of each other.
Solve the given problems. Write the reciprocal of 3 − j in rectangular form.
Perform the indicated operations. Leave the result in polar form.[3(cos32° + j sin32°)][5(cos52° + j sin 52°)]
Solve the given problems. Divide 2 − 3j by its conjugate.
Give the rectangular form of each number:(13.6e2.158j)(3.27e3.888j)
Solve the given problems.Multiply −3 + j by its conjugate.
Give the rectangular form of each number:(35.37e1.096j)2
Using the results of Example 5, find the cube roots of −125.Data from Example 5Find the cube root of −1. Because −1 is a real number, we can find its cube root by means of the definition. Since
Perform the indicated operations and simplify each complex number to its rectangular form. - 3+√18 6
What is the product of the solutions to the equation in Exercise 45?Data from Exercises 45What is the sum of the solutions for the equation x2 − 4x + 13 = 0?
Give the rectangular form of each number:e−3.62j
Find all of the roots of the given equations.x6 + 8 = 0
Perform the indicated operations and simplify each complex number to its rectangular form.−7(2j)(−j3)
What is the sum of the solutions for the equation x2 − 4x + 13 = 0?
Give the rectangular form of each number:2.00e0.25j
Find all of the roots of the given equations.x5 + 32 = 0
Perform the indicated operations and simplify each complex number to its rectangular form.5j(−3j)(j2)
The current in a certain microprocessor circuit is represented by 3.75 ∠15.0° μA. Write this in rectangular form.
Solve the given problems. Show that 1 − j√3 is a solution to the equation x2 + 4 = 2x.
Find all of the roots of the given equations.x4 − j = 0
Give the rectangular form of each number:1.689∠194.36°
Perform the indicated operations and simplify each complex number to its rectangular form.(2√2)2 − j6
The electric field intensity of a light wave can be described by 12.4 ∠ 78.3°V/m. Write this in rectangular form.
Show that −1 − j is a solution to the equation x2 + 2x + 2 = 0.
Give the rectangular form of each number:27.08∠346.27°
Find all of the roots of the given equations.x3 + 27j = 0

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