Math Problem Statement

simplify: 2 + sqrt -8, - sqrt -98, sqrt -216, - sqrt -36, i sqrt -32, sqrt -128, (3i)(-4i)(7i), (-3i)-(3i), (2i)(-8i), (-4i)+(3i), -8i \cdot 4i \cdot i, (8i)^2, i^67, i^45, i^14, i^39300, i^8940, i^50, (-4 - 7i) - (-6+4i), (-5 - 4i) + (-3 + 2i), (4 + 5i)(-7 - 2i) + (1 - 5i) - (-1 + 7i), (-2i)(-5 + 3i)(3 - 8i), 2 sqrt -3 \cdot sqrt -12, (8+i)^2, (-4 + 5i)(-5 + 5i), (-6i)(-7 + 8i) + (i)(-i), -6 (-4i) + (2i)(8 + 5i), sqrt -10 \cdotsqrt -12, (-8 + 7i) + (-4 + 8i) - (1 + 7i)

Solution

This is a collection of complex number expressions that involve both real and imaginary numbers. Let's go through each one and simplify them.

  1. 2+82 + \sqrt{-8}:
    8=8i=22i\sqrt{-8} = \sqrt{8} \cdot i = 2\sqrt{2}i, so: 2+8=2+22i2 + \sqrt{-8} = 2 + 2\sqrt{2}i

  2. 98-\sqrt{-98}:
    98=98i=72i\sqrt{-98} = \sqrt{98} \cdot i = 7\sqrt{2}i, so: 98=72i-\sqrt{-98} = -7\sqrt{2}i

  3. 216\sqrt{-216}:
    216=216i=66i\sqrt{-216} = \sqrt{216} \cdot i = 6\sqrt{6}i, so: 216=66i\sqrt{-216} = 6\sqrt{6}i

  4. 36-\sqrt{-36}:
    36=6i\sqrt{-36} = 6i, so: 36=6i-\sqrt{-36} = -6i

  5. i32i \sqrt{-32}:
    32=32i=42i\sqrt{-32} = \sqrt{32} \cdot i = 4\sqrt{2}i, so: i32=i42i=42i \sqrt{-32} = i \cdot 4\sqrt{2}i = -4\sqrt{2}

  6. 128\sqrt{-128}:
    128=128i=82i\sqrt{-128} = \sqrt{128} \cdot i = 8\sqrt{2}i, so: 128=82i\sqrt{-128} = 8\sqrt{2}i

  7. (3i)(4i)(7i)(3i)(-4i)(7i):
    Multiply the imaginary numbers: (3i)(4i)(7i)=(347)i3=84(i)=84i(3i)(-4i)(7i) = (3 \cdot -4 \cdot 7) \cdot i^3 = -84 \cdot (-i) = 84i

  8. (3i)(3i)(-3i) - (3i):
    (3i)(3i)=6i(-3i) - (3i) = -6i

  9. (2i)(8i)(2i)(-8i):
    (2i)(8i)=16i2=16(2i)(-8i) = -16i^2 = 16

  10. (4i)+(3i)(-4i) + (3i):
    (4i)+(3i)=i(-4i) + (3i) = -i

  11. 8i4ii-8i \cdot 4i \cdot i:
    (8i)(4i)(i)=32i3=32i(-8i)(4i)(i) = -32i^3 = 32i

  12. (8i)2(8i)^2:
    (8i)2=64i2=64(8i)^2 = 64i^2 = -64

  13. i67i^{67}:
    Since i4=1i^4 = 1, we use i67=i67mod4=i3=ii^{67} = i^{67 \mod 4} = i^3 = -i.

  14. i45i^{45}:
    i45=i45mod4=ii^{45} = i^{45 \mod 4} = i.

  15. i14i^{14}:
    i14=i14mod4=i2=1i^{14} = i^{14 \mod 4} = i^2 = -1.

  16. i39300i^{39300}:
    i39300=i39300mod4=i0=1i^{39300} = i^{39300 \mod 4} = i^0 = 1.

  17. i8940i^{8940}:
    i8940=i8940mod4=i0=1i^{8940} = i^{8940 \mod 4} = i^0 = 1.

  18. i50i^{50}:
    i50=i50mod4=i2=1i^{50} = i^{50 \mod 4} = i^2 = -1.

  19. (47i)(6+4i)(-4 - 7i) - (-6 + 4i):
    Subtract complex numbers: (47i)(6+4i)=(4+6)+(7i4i)=211i(-4 - 7i) - (-6 + 4i) = (-4 + 6) + (-7i - 4i) = 2 - 11i

  20. (54i)+(3+2i)(-5 - 4i) + (-3 + 2i):
    (54i)+(3+2i)=(53)+(4i+2i)=82i(-5 - 4i) + (-3 + 2i) = (-5 - 3) + (-4i + 2i) = -8 - 2i

  21. (4+5i)(72i)+(15i)(1+7i)(4 + 5i)(-7 - 2i) + (1 - 5i) - (-1 + 7i):
    First, multiply (4+5i)(72i)(4 + 5i)(-7 - 2i): [ (4 + 5i)(-7 - 2i) = -28 -

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Imaginary Numbers
Operations on Complex Numbers

Formulas

i^2 = -1
√-x = i√x
Multiplication and addition of complex numbers
Power reduction for imaginary units i^n

Theorems

Imaginary Unit Cyclic Powers (i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1)

Suitable Grade Level

Grades 10-12