Recall that \(a^2 - b^2 = (a-b)(a+b)\)
Can you take it from here?
Let \(x = \sqrt{-16 + 31i}\) and be in the form a + bi.
We have: \((a+bi)^2 = (a+bi)(a+bi) = a^2 + 2abi - b^2 \)
We know that \(2abi = 31\), meaning \(ab = {31 \over 2}\)
Now, remember that \(a^2 - b^2 = -16\)
This gives us the system:
\(a^2 - b^2 = -16\) (i)
\(ab = {31 \over 2}\) (ii)
Using WA, we find the answer to be: \(\sqrt{-16 + 31i} = \color{brown}\boxed{\sqrt{{\sqrt{1217}\over 2} - 8} + \sqrt{8 + {\sqrt{1217}\over 2}}i}\)
Note that \({{x^2 + 1} \over x} = {x^2 \over x} + {1 \over x} = x + {1 \over x}\)
Now, simplify the equation to: \(3x + {2 \over x} - 4 = 2x -5+ {1 \over x}\)
Now, subtract \(2x \) from both sides: \(x + {2 \over x} - 4 = -5 + {1 \over x}\)
Subtract \({1 \over x}\) from both sides: \(x + {1 \over x} - 4 = -5\)
The formula for the number of diagonals in a polygon is \(s(s-3) \div 2\), where s is the numberof sides.
Setting this equal to 2s, we have: \({s(s-3) \over 2} = 2s\)
Solving, we find s = 0 or 7.
But, s can't be 0, so \(s = \color{brown}\boxed{7}\)
Take \(ab + c = 2\). From the first words, we know that c must be 1, because if c was 2, \(ab = 0\), which is not possible with positive integers.
This means \(c = 1 \), and \(ab = 1\).
The only positive integers that satisfy this are: \(a = b = c = 1 \).
Thus, \(a + b + c = \color{brown}\boxed{3}\)
https://web2.0calc.com/questions/find-a-base-7-three-digit-number-which-has-its-digits-reversed-when-expressed-in-base-9-you-do-not-need-to-indicate-the-base-with-a-subscr
Wish I found this link before spending 30 min on this....
The equation factors to: \((m+4)(n-1) = 36\)
Now, recall the factor of 36: 1,2,3,4,6,9,12,18,36
We need to find the number of pairs of integers that satisfy this.
For example, 1 pair is \((32,2)\), because 36 x 1 = 36.
Note that the points have the same y-coordinate, which is 6.
This means that the line is \(\color{brown}\boxed{y = 6}\)
Simplify to: \(5x + 5 = 25\)
Subtract 5 from both sides: \(5x = 20\)
Now, tell me, what number multiplied by 5 is 20.
Note that the exterior angles will always sum to \(360^ \circ\).
The ratio also has 15 "parts", meaning each part is \(360 \div 15 = 24\)
The smallest interior angle will always be supplementary to the largest exterior angle, which is \(7 \times 24 = 168\).
This means that the smallest interior angle is \(180 - 168 = \color{brown}\boxed{12}\)