)SIMPLIFY i^2 square root of -80

2)SOLVE 49x^2+100=0

3)14-i^6 divided by i

1) To simplify the expression i^2 * √(-80), we need to simplify each component of the expression separately.

First, let's simplify i^2. The square of the imaginary unit i is -1, so i^2 = -1.

Next, let's simplify the square root of -80. We can break down -80 as -1 * 80. Taking the square root of -1 gives us i, and the square root of 80 can be simplified as 4√5.

Putting everything together, i^2 * √(-80) becomes (-1) * (i * 4√5) = -4i√5.

Therefore, the simplified form of i^2 * √(-80) is -4i√5.

2) To solve the equation 49x^2 + 100 = 0, we can start by subtracting 100 from both sides of the equation to isolate the term with x^2:

49x^2 = -100

Next, divide both sides of the equation by 49 to solve for x^2:

x^2 = -100/49

To find the value of x, we need to take the square root of both sides of the equation. However, it's important to note that the square root of a negative number is not a real number. The square root of a negative number can be represented using the imaginary unit i.

The square root of -1 is denoted as i, so we can write the square root of -100/49 as ±(√100/49)i.

Simplifying this expression, we get:

x = ±(10/7)i

Therefore, the solutions to the equation 49x^2 + 100 = 0 are x = (10/7)i and x = -(10/7)i.

3) To simplify (14 - i^6) / i, we need to simplify each component of the expression separately.

First, let's simplify i^6. The imaginary unit i raised to the power of 6 is equal to (i^2)^3. Since i^2 is equal to -1, i^6 becomes (-1)^3, which is equal to -1.

Now, let's simplify the expression (14 - i^6). Since i^6 is -1, we can substitute it in:

(14 - i^6) = (14 - (-1)) = 15.

Finally, dividing 15 by i gives us:

(14 - i^6) / i = 15 / i.

To simplify further, we can multiply the numerator and denominator by -i to get rid of the fraction:

(15 / i) * (-i / -i) = -15i / i^2.

Since i^2 is equal to -1, we can simplify further:

-15i / i^2 = -15i / (-1) = 15i.

Therefore, (14 - i^6) / i simplifies to 15i.