Math Problem Statement

Fully simplify \((-9\sqrt{-42})(5\sqrt{-6})\).

Solution

To fully simplify the expression (942)(56)( -9\sqrt{-42})(5\sqrt{-6}), we need to handle the square roots of negative numbers, which involve imaginary numbers. Here’s a step-by-step simplification:

  1. Rewrite the square roots of negative numbers: 42=42iand6=6i\sqrt{-42} = \sqrt{42} \cdot i \quad \text{and} \quad \sqrt{-6} = \sqrt{6} \cdot i where ii is the imaginary unit, defined as i=1i = \sqrt{-1}.

  2. Substitute and expand the expression: (942)(56)=(9)(5)(42i)(6i)(-9\sqrt{-42})(5\sqrt{-6}) = (-9)(5)(\sqrt{42} \cdot i)(\sqrt{6} \cdot i)

  3. Multiply the constants and imaginary terms: =(95)(426)i2= (-9 \cdot 5) \cdot (\sqrt{42} \cdot \sqrt{6}) \cdot i^2 = -45 \cdot \sqrt{42 \cdot 6} \cdot (-1) \quad \text{(since i^2 = -1)}

  4. Calculate the product under the square root: 426=252\sqrt{42 \cdot 6} = \sqrt{252}

  5. Simplify 252\sqrt{252}: 252=463=263=297=237=67\sqrt{252} = \sqrt{4 \cdot 63} = 2\sqrt{63} = 2\sqrt{9 \cdot 7} = 2 \cdot 3 \cdot \sqrt{7} = 6\sqrt{7}

  6. Combine everything: 4567(1)=2707-45 \cdot 6\sqrt{7} \cdot (-1) = 270\sqrt{7}

So, the fully simplified expression is: 2707270\sqrt{7}

Would you like further clarification on any step, or do you have additional questions?

Here are some related questions to explore further:

  1. What are imaginary numbers, and why do they involve ii?
  2. How do you simplify expressions involving square roots of negative numbers?
  3. What is the process to simplify square roots like 252\sqrt{252}?
  4. How would the expression change if it had additional imaginary terms?
  5. How can you apply similar steps to other expressions with complex numbers?

Tip: When dealing with square roots of negative numbers, always consider rewriting them in terms of ii (imaginary unit) to simplify further.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Imaginary Numbers
Simplification of Radical Expressions

Formulas

\( i = \sqrt{-1} \)
\( i^2 = -1 \)

Theorems

Properties of Imaginary and Complex Numbers

Suitable Grade Level

Grades 10-12