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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Perform the indicated operations and simplify each complex number to its rectangular form.(√−2)2 + j4
Find the magnitude and direction of a force on a bolt that is represented by 40.5 + 24.5 j newtons.
Evaluate each expression on a calculator. Express answers in the form a + bj. (2-j³)4 (j8 - j6)³ + j
Evaluate each expression on a calculator. Express answers in the form a + bj.(1 + j)−3(2 − j)−2
Give the rectangular form of each number:20∠160°
Find all of the roots of the given equations.x3 − 8 = 0
Perform the indicated operations and simplify each complex number to its rectangular form.√−27 + √12
The voltage of a certain generator is represented by 2.84 − 1.06j kV. Write this voltage in polar form.
Give the rectangular form of each number:0.62∠−72°
Find all of the roots of the given equations.x4 − 1 = 0
Perform the indicated operations and simplify each complex number to its rectangular form.√18 −√−8
The impedance in a certain circuit is Z = 8.5∠−36°Ω. Write this in rectangular form.
Evaluate each expression on a calculator. Express answers in the form a + bj.(5j − 4j2 + 3 j7)(2j12 − j13)
If E = 115e0.315j V and I = 28.6e−0.723j A, find the exponential form of Z given that E = IZ.
Give the rectangular form of each number:2.417(cos656.26° + j sin656.26°)
Use DeMoivre’s theorem to find all the indicated roots. Be sure to find all roots.The cube roots of √3 + j
Perform the indicated operations and simplify each complex number to its rectangular form.j3 − 6
Solve the given problems. Show that the conjugate of r ∠ θ is r ∠−θ.
In an electric circuit, the admittance is the reciprocal of the impedance. If the impedance is 2800 − 1450j ohms in a certain circuit, find the exponential form of the admittance.
Give the rectangular form of each number:5.011(cos123.82° + j sin123.82°)
Use DeMoivre’s theorem to find all the indicated roots. Be sure to find all roots.The square roots of 1 + j
Perform the indicated operations and simplify each complex number to its rectangular form.2j2 + 3j
Relative to the air, a plane heads north of west with a velocity that can be represented by −480 + 210j km/h. The wind is blowing from south of west with a velocity that can be represented by 60 +
Perform the indicated operations and simplify each complex number to its rectangular form.5 − 2√25j2
For x + yj, what is the argument if x = y < 0?
The intensity of the signal from a radar microwave signal is 37.0[cos(−65.3°) + j sin(−65.3°)] V/m. Write this in exponential form.
Give the rectangular form of each number:48(cos60° + j sin60°)
Perform the indicated operations, expressing all answers in the form a + bj.(2j2 − 3j3 + 2j4 − 2j5)6
Use DeMoivre’s theorem to find all the indicated roots. Be sure to find all roots.The two square roots of −5 + 12j
Perform the indicated vector additions graphically. Check them algebraically.Two ropes hold a boat at a dock. The tensions in the ropes can be represented by 40 + 10j lb and 50 − 25j lb. Find the
Perform the indicated operations and simplify each complex number to its rectangular form.√−4j2 + √−4
What is the argument for any negative real number?
Perform the indicated operations.The impedance in an antenna circuit is 3.75 + 1.10j ohms. Write this in exponential form and find the magnitude of the impedance.
Give the rectangular form of each number:2(cos225° + j sin 225°)
Perform the indicated operations, expressing all answers in the form a + bj.(4j5 − 5j4 + 2j3 − 3j2)2
Use DeMoivre’s theorem to find all the indicated roots. Be sure to find all roots.The three cube roots of 3 − 4j
Using a calculator, express 75.6e1.25j in rectangular form. See Fig. 12.15. NORMAL FIX3 AUTO REAL RADIAN MP 2-4 Polar Ans Rect 4.472e 1.107 0 2.000-4.0001 Fig. 12.15
Perform the indicated operations and simplify each complex number to its rectangular form.−√1 −√−400
Perform the indicated vector operations graphically on the complex number 2 + 4j.Graph the number, the number multiplied by j, the number multiplied by j2, and the number multiplied by j3 on the same
Represent each complex number graphically and give the rectangular form of each.4629∠182.44°
Give the polar and exponential forms of each of the complex numbers.−4j5
Perform the indicated operations, expressing all answers in the form a + bj. (6j+5)(24j) (5 - j)(4j + 1)
Use DeMoivre’s theorem to find all the indicated roots. Be sure to find all roots. The three cube roots of 27(cos 120° + j sin 120°)
Perform the indicated vector operations graphically on the complex number 2 + 4j. Subtract the conjugate from the number. Describe the result.
Perform the indicated operations and simplify each complex number to its rectangular form.3j −√−100
Using a calculator, express 3.73 + 5.24j in exponential form. See Fig. 12.15. NORMAL FIX3 AUTO REAL RADIAN MP 2-4 Polar Ans Rect 4.472e 1.107 0 2.000-4.0001 Fig. 12.15
Represent each complex number graphically and give the rectangular form of each.86.42∠94.62°
Perform the indicated operations, expressing all answers in the form a + bj. 4j 1-j j + 8 2+ 3j
Give the polar and exponential forms of each of the complex numbers.5000
Perform the indicated operations, expressing all answers in the form a + bj. 3 2j 5 j - 6
Use DeMoivre’s theorem to find all the indicated roots. Be sure to find all roots.The two square roots of 4(cos60° + j sin60°)
Perform the indicated vector operations graphically on the complex number 2 + 4j.Add the number and its conjugate. Describe the result.
Perform the indicated operations and simplify each complex number to its rectangular form.−26 + √−64
Represent each complex number graphically and give the rectangular form of each.18.3∠540.0°
Give the polar and exponential forms of each of the complex numbers.158j − 327
Perform the indicated operations by using properties of exponents and express results in rectangular and polar forms.(18.0e5.13j)(25.5e0.77j)
Perform the indicated vector operations graphically on the complex number 2 + 4j.Graph the complex number and its conjugate. Describe the relative positions.
Perform the indicated operations by using properties of exponents and express results in rectangular and polar forms.(625e3.46j)(4.40e1.22j)
Give the polar and exponential forms of each of the complex numbers.1.07 + 4.55j
Represent each complex number graphically and give the rectangular form of each.7.32∠-270°
Perform the indicated operations and simplify each complex number to its rectangular form.2 + √−9
Perform the indicated operations, expressing all answers in the form a + bj. j² - j 2j - j8
Change each number to polar form and then perform the indicated operations. Express the result in rectangular and polar forms. Check by performing the same operation in rectangular form.(3 + 4j)4
Show the numbers a + bj, 3(a + bj), and −3(a + bj) on the same coordinate system. The multiplication of a complex number by a real number is called scalar multiplication of the complex number.−10
Perform the indicated operations, expressing all answers in the form a + bj. js- j³ 3 + j
Simplify each of the given expressions.−√−j2
Represent each complex number graphically and give the rectangular form of each.277.8∠-342.63°
Give the polar and exponential forms of each of the complex numbers.60 − 20j
Perform the indicated operations by using properties of exponents and express results in rectangular and polar forms.(0.926e0.253j)3
Change each number to polar form and then perform the indicated operations. Express the result in rectangular and polar forms. Check by performing the same operation in rectangular form. 5j - 2 -1-j
Perform the indicated operations on the resulting expressions if the given changes are made in the indicated examples of this section.In Example 2(b), change the sign before 6.2j from − to +, and
Locate the given numbers in the complex plane.2 + 6j
Perform the indicated operations if the given changes are made in the indicated examples of this section.In Example 5, double the values of L and C and then solve the given problem.Data from Example
Perform the indicated operations for the resulting complex numbers if the given changes are made in the indicated examples of this section.In Example 2, change the sign of the angle in the second
Perform the indicated operations for the resulting complex numbers if the given changes are made in the indicated examples of this section.In Example 3, change the exponent to 3.80j and then find the
Determine each of the following as being either true or false. If it is false, explain why.4/90° = 4j
Represent each complex number graphically and give the polar form of each.8 + 6j
In Example 5(a), add 40 to the exponent and then evaluate.Data from Example 5(a)j10 = j8j2 = (1)(−1) = −1
Express the given numbers in exponential form.3.00(cos 60.0° + j sin 60.0°)
Multiply, expressing the result in polar form: (2∠130°)(3∠45°).
Perform the indicated operations for the resulting complex numbers if the given changes are made in the indicated examples of this section.In Example 1, change 136.3° to 226.3° and then find the
Perform the indicated operations if the given changes are made in the indicated examples of this section.In Example 1, change the value of XL to 16.0Ωand then solve the given problem.Data from
In Example 4(a), put a j in front of the radical and then simplify.Data from Example 4(a) √-6 = √(6)(-1) = √6√-1 = j√6 this step is correct if only one is negative, as in this case
Add, expressing the result in rectangular form: (3 −√−4) + (5√−9 − 1).
Perform the indicated operations for the resulting complex numbers if the given changes are made in the indicated examples of this section.In Example 3, change the sign of the imaginary part of the
Change the sign of the real part of the complex number in the indicated example of this section and then perform the indicated operations for the resulting complex number.Example 3Data from Example
Determine each of the following as being either true or false. If it is false, explain it why. 1+ + j 1-j j
Perform the indicated operations on the resulting expressions if the given changes are made in the indicated examples of this section.In Example 1(b), change the sign in the first parentheses from +
Perform the indicated operations for the resulting complex numbers if the given changes are made in the indicated examples of this section.In Example 1, change the sign of the angle in the first
Change the sign of the real part of the complex number in the indicated example of this section and then perform the indicated operations for the resulting complex number.Example 1Data from Example
Perform the indicated operations for the resulting complex numbers if the given changes are made in the indicated examples of this section.In Example 2(a), change the sign of the imaginary part of
In Example 3, put a − sign before the first radical of the first illustration and then simplify.Data from Example 3To further illustrate the method of handling square roots of negative numbers,
A computer analysis of an experiment showed that the fraction f of viruses surviving X-ray dosages was given bywhere d is the dosage. Express this with the denominator rationalized. f = 20 d+√3d +
Find the combined impedance (in rectangular form) of the circuit elements in Fig. 12.31. The frequency of the current in the circuit is 60.0 Hz. See Exercise 22.Data from Exercises 22Find the
Determine each of the following as being either true or false. If it is false, explain why.(√−9)2 = 9
Perform the indicated operations. Express results in polar form. See Example 3.15.9∠142.6° − 18.5∠71.4°Data from Example 3Perform the addition 1.563∠37.56° + 3.827∠146.23°. In order to
Simplify each of the given expressions. 49 -√(-4)(-48) 16
In calculating the forces on a tower by the wind, it is necessary to evaluate 0.0180.13. Write a paragraph explaining why this form is preferable to the equivalent radical form.
Perform the indicated operations. Express results in polar form. See Example 3.2.78∠56.8° + 1.37∠207.3°Data from Example 3Perform the addition 1.563∠37.56° + 3.827∠146.23°. In order to do
For two impedances Z1 and Z2 in parallel, the combined impedance ZC is given byFind the combined impedance (in rectangular form) for the parallel circuit elements in Fig. 12.30 if the current in the

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