Question

as an imaginary nu

1. √(-81)

of i.

Explanation:

To solve the expression \( 2 \cdot \sqrt{-81} \) (assuming that's the intended expression, as we work with imaginary numbers here), we can follow these steps:

Step 1: Recall the definition of the imaginary unit \( i \)

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \). This allows us to rewrite square roots of negative numbers in terms of \( i \).

Step 2: Simplify \( \sqrt{-81} \)

We can factor out the negative sign inside the square root:

$$ \sqrt{-81} = \sqrt{81 \cdot (-1)} $$

Using the property of square roots \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \) (valid for real numbers \( a \geq 0, b \geq 0 \); here we extend it to include negative numbers by using \( i \)):

$$ \sqrt{81 \cdot (-1)} = \sqrt{81} \cdot \sqrt{-1} $$

We know \( \sqrt{81} = 9 \) and \( \sqrt{-1} = i \), so:

$$ \sqrt{-81} = 9i $$

Step 3: Multiply by the coefficient \( 2 \)

Now substitute \( \sqrt{-81} = 9i \) back into the original expression \( 2 \cdot \sqrt{-81} \):

$$ 2 \cdot \sqrt{-81} = 2 \cdot 9i $$

Step 4: Simplify the product

Multiply \( 2 \) and \( 9 \):

$$ 2 \cdot 9i = 18i $$

So, \( 2 \cdot \sqrt{-81} = \boldsymbol{18i} \).

Answer:

To solve the expression \( 2 \cdot \sqrt{-81} \) (assuming that's the intended expression, as we work with imaginary numbers here), we can follow these steps:

Step 1: Recall the definition of the imaginary unit \( i \)

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \). This allows us to rewrite square roots of negative numbers in terms of \( i \).

Step 2: Simplify \( \sqrt{-81} \)

We can factor out the negative sign inside the square root:

$$ \sqrt{-81} = \sqrt{81 \cdot (-1)} $$

Using the property of square roots \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \) (valid for real numbers \( a \geq 0, b \geq 0 \); here we extend it to include negative numbers by using \( i \)):

$$ \sqrt{81 \cdot (-1)} = \sqrt{81} \cdot \sqrt{-1} $$

We know \( \sqrt{81} = 9 \) and \( \sqrt{-1} = i \), so:

$$ \sqrt{-81} = 9i $$

Step 3: Multiply by the coefficient \( 2 \)

Now substitute \( \sqrt{-81} = 9i \) back into the original expression \( 2 \cdot \sqrt{-81} \):

$$ 2 \cdot \sqrt{-81} = 2 \cdot 9i $$

Step 4: Simplify the product

Multiply \( 2 \) and \( 9 \):

$$ 2 \cdot 9i = 18i $$

So, \( 2 \cdot \sqrt{-81} = \boldsymbol{18i} \).