×

Hello! Socratic's Terms of Service and Privacy Policy have been updated, which will be automatically effective on October 6, 2018. Please contact hello@socratic.com with any questions.

# How do you simplify (7sqrt(-3))(2sqrt(-27))?

Mar 8, 2018

#### Answer:

Always deal with imaginary numbers first. Final answer: $- 126$

#### Explanation:

If you ignore the rule of imaginary numbers which states that $\sqrt{- 1} =$$i$ and ${i}^{2}$=-1, you would be tempted to multiply all terms together to get $14 \sqrt{81} = 14 \cdot 9 = 126$ ... which would be incorrect.

To keep the validity of the complex number system, always deal with the imaginary part first. To do this, rewrite the equation:

$\left(7 \sqrt{3} \cdot \sqrt{- 1}\right) \left(2 \sqrt{27} \cdot \sqrt{- 1}\right) = \left(\sqrt{- 1} \cdot \sqrt{- 1}\right) \left(7 \sqrt{3} \cdot 2 \sqrt{27}\right)$

Now you can multiply terms within the parentheses.

$\left(\sqrt{- 1} \cdot \sqrt{- 1}\right) = i \cdot i = {i}^{2} = - 1$

and

$\left(7 \sqrt{3} \cdot 2 \sqrt{27}\right) = 14 \sqrt{81} = 14 \cdot 9 = 126$

Multiply these together and you get $126 \cdot - 1 = - 126$