Download Radicals and Complex Numbers

 Regents Review Session #2
Radicals, Imaginary Numbers and Complex Numbers What do you do to simplify radicals?
1. Break the radical into two radicals ­ one that is a perfect square and one that is the other factor.
2. Simplify the perfect square radical and leave the non­perfect square radical.
** Remember, if there is a variable, under a square root, even powers on the variable are perfect squares. If it is a cube root, variables raised to a multiple of 3 are perfect cubes, etc.***
1. √24x12y9
2. ∛128x6y10
3. ∜243x9y12
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What to do when you have to add/subtract radicals?
1. You must have like terms (the same radicands) to add/subtract.
2. Simplify the radicals and add/subtract (combine the coefficients and keep the radical) if possible.
1. 2√108 + 4√27 - 9√50
What do you when you when you multiply radicals together?
1. You multiply like parts, coefficient times coefficient and radicand times radicand.
2. Simplify when necessary.
Examples:
1. ﴾7 ­ 2√3﴿2
2. ﴾5√2 ­ 3√6﴿﴾√2 + 4√6﴿
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Rationalizing Denominators
What do you do when you have two terms in the denominator and one
of them is a radical?
1. Multiply the entire fraction by the conjugate of the denominator.
Remember, when you multiply conjugates together, you the outer and
the inner terms will always cancel out, so you only need to multiply
‘firsts’ and ‘lasts’
2. Simplify when possible.
Examples:
1. Write an expression that is equal to 2 + √5
2 - √5
2. √6 + 4√3
√6 ­ 8√3
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Solving Radical Equations
What do you do when you have to solve a radical equation?
1. You must isolate the radical term first.
2. Square both sides.
3. Solve for x.
4. Check your solutions; you need to ensure you do not have extraneous solutions.
Examples:
Solve for x.
1.
√(x + 2﴿ + x = 10
2. 1 + √2x =√3x + 1
3. The equation V = 20√﴾C + 273﴿ relates speed of sound, V, in meters per second, to air temperature, C, in degrees Celsius. What is the temperature, in degrees Celsius, when the speed of sound is 320 meters per second?
4. The lateral surface area of a right circular cone, s, is represented by the equation s = πr√﴾r + h2﴿
where r is the radius of the circular base and h is the height of the cone. If the lateral surface area of a large funnel is 236.64 square centimeters and its radius is 4.75 centimeters, find its height, to the nearest hundredth of a centimeter
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What do you do when you have to simplify the square root of a negative number?
1. You must take the negative out from under the radical first!!!
2. Simplify further if possible.
Examples:
1. √­4
2. √­24
3. ­ ½ √­80
4. √­100 + √­81
5. 8 + √­28 ­ 7 + √­6
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What do you do when you have to simplify powers of i?
1. Remember i1 = i
i2 = ­1
i3 = ­i
i4 = 1 and the pattern of i repeats in the cycle of 4.
2. Divide the exponent by four, the remainder is the new exponent on i.
3. Simplify that power of i.
Examples:
1. i35
5.
2. i72
4i + 5i8 + 6i3 + 2i4
3. i49 4. i4n + 2 6. i + i2 + i3 + i4
7. i * i2 * i3 * i4
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Complex Numbers
What do you do when you have to add or subtract complex numbers?
1. You add or subtract the like parts, real plus/minus real, imaginary plus/minus imaginary.
Examples:
1. ﴾3 + 2i﴿ + ﴾5 ­ 9i﴿
2. ﴾­2 + 5i﴿ ­ ﴾7 ­ 3i﴿
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What do you do when you have to multiply complex numbers?
1. A complex number times a complex number requires that you FOIL/distribute. 2. Simplify powers of i.
3. Combine like terms.
Examples:
2. ﴾7 ­ 2i﴿﴾3 + 4i﴿
1. ﴾2 + 3i﴿﴾2 ­ 3i﴿
3. ﴾4√2 ­ 3i﴿﴾9√2 + 4i﴿
4. ﴾2 + 5i﴿2
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What do you do when you have two complex numbers set equal to each other?
1. Set the real parts equal to each other.
2. Set the imaginary coefficients equal to each other.
3. Solve each equation separately.
Examples:
1. Find x and y.
2. Find x and y.
﴾x + 2﴿ ­ 8i = 15 + ﴾2y­10﴿i
﴾6x ­ 2﴿ + 9i = 7 ­ ﴾y­4﴿i
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What do you do when you divide by a complex number?
1. You must multiply the numerator and denominator by the complex conjugate of the denominator.
2. Distribute/foil, simplify powers of i, and combine like terms. 3. If asked to write your answer in simplest a + bi form, separate the fractions.
Examples: Simplify and write your answer in simplest a + bi form.
1. ­4 2. 5 3. 5 + 5i
5 ­ 3i 2 + i
3 ­ 4i
4. 2 ­ 4i√3
3 + i√3
5. Find the multiplicative inverse of 4 ­ 2i
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What do you do when you have to solve a quadratic equation with complex roots?
1. Make sure the quadratic equation is set equal to zero. 2. Use the quadratic formula to solve for x.
3. Simplify the radical/imaginary term and write in simplest a + bi form.
Examples:
Use the quadratic formula to solve for x and state your answer in simplest a + bi form.
2. 3x2 ­ 5x = ­8
1. 9x + 2/x = ­6
3. ­ x2 + 3x ­ 12 = 0
4. x2 + 1 = 4(x ­ 1)
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What do you do when you have to graph complex numbers?
1. To graph a complex number, you graph the point (a, b) then connect the point to the origin with a directed segment. Examples:
1. Given z1 = 4 + 3i and z2 = ­7 ­ i,
a. Graph z1 and z2.
b. What is the value of z1 + z2?
c. In what quadrant does the sum of z1 + z2 ­ lie?
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2.
Two complex numbers are graphed below. What is the sum of w and u, expressed in standard complex number form?
﴾1﴿ 7 + 3i ﴾2﴿ 5 + 7i
﴾3﴿ 3 + 7i ﴾4﴿ ­5 + 3i
3. When the sum of ­11 + 5i and ­3 + 9i is graphed, in which quadrant does it lie?
4. If z 1 = ­2 + 7i and z2 = 6 – 5i in which quadrant does the graph of z2 ­ z1 lie?
5. On a graph, if point A represents ­8 + 9i and point B represents 1­ 2i which quadrant contains 3A ­ 2B?
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