Math Problem Statement

-8 positive negative square root of -27 all over 6

Solution

To simplify the expression 8±27/6-8 \pm \sqrt{-27} \, \bigg/ \, 6:

  1. Simplify the square root of 27-27:

    • 27\sqrt{-27} can be written as 1×27=1×27=i×27\sqrt{-1 \times 27} = \sqrt{-1} \times \sqrt{27} = i \times \sqrt{27}.
    • Since 27\sqrt{27} simplifies to 333\sqrt{3}, we get 27=3i3\sqrt{-27} = 3i\sqrt{3}.
  2. Rewrite the expression: 8±3i36\frac{-8 \pm 3i\sqrt{3}}{6}

  3. Simplify the fraction:

    • Each term in the numerator can be divided by the denominator: 86±3i36\frac{-8}{6} \pm \frac{3i\sqrt{3}}{6}
    • Simplify both fractions: 43±i32-\frac{4}{3} \pm \frac{i\sqrt{3}}{2}

So, the simplified expression is: 43±i32-\frac{4}{3} \pm \frac{i\sqrt{3}}{2}

Would you like more details or have any questions?

Here are 5 related questions for further practice:

  1. Simplify 6±504\frac{6 \pm \sqrt{-50}}{4}.
  2. What are the solutions for the quadratic equation x2+4x+13=0x^2 + 4x + 13 = 0?
  3. Simplify the expression 16\sqrt{-16}.
  4. How would you simplify 3±122\frac{-3 \pm \sqrt{-12}}{2}?
  5. Find the roots of the equation 2x2+3x+5=02x^2 + 3x + 5 = 0.

Tip: When dealing with square roots of negative numbers, remember that 1=i\sqrt{-1} = i, which is the imaginary unit.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Square Roots
Imaginary Numbers

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12