1)SIMPLIFY i^2 square root of -80
2)SOLVE 49x^2+100=0
3)14-i^6 divided by i
1. i^2sqrt(-80),
i^2*sqrt(16*5*-1).
-1*sqrt(16*5*-1),
-1 * 4i * sqrt(5),
-4i * sqrt(5).
2.49x^2 + 100 = 0.
3. (14 - i^6) / i ,
(14 - i^2*i^2*i^2) / i,
(14 - (-1)(-1)(-1)) / i,
(14 - (-1)) / i,
(14 + 1) / i,
15 / i,
Multiply numerator and denominator by
i:
15i / i^2,
15i / -1 = - 15i.
1) To simplify i^2√(-80), we can start by simplifying the square root of -80.
The first step is to express -80 as the product of a perfect square and a negative number. We know that the perfect square of 8 is 64. Therefore, we can rewrite -80 as -1 * 64 * 5.
Now we can simplify the square root by taking the square root of the perfect square and placing it outside the square root, i.e., i√(-1 * 64 * 5). The square root of 64 is 8, so we get 8i√(-5).
Therefore, i^2√(-80) simplifies to 8i√(-5).
2) To solve the equation 49x^2 + 100 = 0, we can start by subtracting 100 from both sides: 49x^2 = -100.
Then, divide both sides of the equation by 49: x^2 = -100/49.
To remove the square root, we can take the square root of both sides, using the principle that the square root of a square cancels out the square: x = ±√(-100/49).
The expression inside the square root is negative, which means that there are no real solutions to this equation. However, we can express the solutions using complex numbers. So, the solutions to the equation 49x^2 + 100 = 0 are x = ±i√(100/49).
3) To simplify (14 - i^6) divided by i, we can start by simplifying i^6.
The powers of i repeat in a cycle: i, -1, -i, 1. Since 6 is divisible by 4, we know that i^6 will be the same as i^4, which is equal to 1.
Therefore, we can rewrite (14 - i^6) divided by i as (14 - 1) divided by i.
Simplifying further, we get 13 divided by i.
To rationalize the denominator, we can multiply the numerator and denominator by -i, which is the complex conjugate of i. This results in:
(13 * -i) divided by (i * -i)
Simplifying this, we get -13i divided by -i^2.
Since i^2 equals -1, we have -13i divided by -(-1).
Simplifying further, we get -13i divided by 1, which is simply -13i.
Therefore, (14 - i^6) divided by i simplifies to -13i.