Math Problem Statement

Simplify the given expression. negative StartRoot negative 3 EndRoot StartRoot negative 5 EndRoot StartRoot negative 15 EndRoot Question content area bottom Part 1 negative StartRoot negative 3 EndRoot StartRoot negative 5 EndRoot StartRoot negative 15 EndRootequals    enter your response here ​(Simplify your answer. Type your answer in the form aplusbj​.)

Solution

Let's simplify the given expression:

3×5×15-\sqrt{-3} \times \sqrt{-5} \times \sqrt{-15}

Step 1: Simplify the square roots involving negative numbers.

We know that the square root of a negative number involves the imaginary unit ii, where:

1=i\sqrt{-1} = i

Thus, we can express each square root with a negative number as:

3=i3,5=i5,15=i15\sqrt{-3} = i\sqrt{3}, \quad \sqrt{-5} = i\sqrt{5}, \quad \sqrt{-15} = i\sqrt{15}

Step 2: Rewrite the expression with imaginary parts.

Substitute these into the expression:

3×5×15=(i3)×(i5)×(i15)-\sqrt{-3} \times \sqrt{-5} \times \sqrt{-15} = -\left(i\sqrt{3}\right) \times \left(i\sqrt{5}\right) \times \left(i\sqrt{15}\right)

Step 3: Simplify the multiplication.

Multiply the constants and imaginary units separately:

(1)×i×i×i=i3=(i)=i(-1) \times i \times i \times i = -i^3 = -(-i) = i

Now multiply the square roots:

3×5×15=3×5×15=225=15\sqrt{3} \times \sqrt{5} \times \sqrt{15} = \sqrt{3 \times 5 \times 15} = \sqrt{225} = 15

Step 4: Combine everything.

The expression simplifies to:

i×15=15ii \times 15 = 15i

Final Answer:

15i15i


Would you like to see more details on any part of the simplification, or have any questions? Here are a few related questions you might find helpful:

  1. How do you simplify expressions involving square roots of negative numbers?
  2. What is the property of ii when multiplying it (e.g., i2i^2, i3i^3)?
  3. How do you handle complex numbers in multiplication?
  4. What happens when you multiply an imaginary number by a real number?
  5. How do you simplify square roots involving variables?

Tip: Always break down complex expressions involving square roots of negative numbers by factoring out ii first, then simplify the real parts separately!

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Square Roots
Imaginary Numbers

Formulas

\sqrt{-1} = i
i^2 = -1
i^3 = -i

Theorems

Multiplication of Complex Numbers
Square Root of Negative Numbers

Suitable Grade Level

Grades 9-12