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\(P = 5^{4/13}\)

Therefore, a + b + c = 5 + 4 + 13 = 22.

(4-5i)(5+5i)=4(5)+4(5i)-5i(5)-5i(5i)=20+20i-25i-25i^2=20-5i-25(-1)=20-5i+25=45-5i is your answer.

A Total of 7?
(3, 7, 8)

(4, 7, 7)

(2, 8, 8)

(6, 6, 6)
(4, 6, 8)

(5, 5,  8)
(5, 6, 7)

PolynomialGCD[10 p^2, 25 p q, 15 q^2, 2 p]==1

Vieta's formula states that the sum of all the roots of a polynomial is \(-{ b \over a} \)

Subbing in what we know, we find that the sum of the roots is \({121 \over 24}\).

Now, we just have to find the logarithm of this. 

Can you take it from here?

The modular inverse of 27, modulo 29 ==14, because:

27 mod 29 ==27,  and:

[27 x 14] mod 29 ==1

There will be 2 distinct roots if and only if the discriminant (\(b^2 - 4ac\)) is positive. 

Substituting what we know, we have: \(m^2 - 4 \times 2 \times 18 > 0\)

Now, we have a simple inequality and have to solve for m.

Can you take it from here?

I tried some values, and got \(x_6 = 8\)

Any other answers? That waas incorrect.

That's not right. I tried it? Do y'all have an explanation forthat?

Acute isosceles triangle.
Sides: a = 6 b = 6 c = 5
Use Heron's Formula to find the area of the triangle: Heron's formula gives the area of a triangle when the length of all three sides is known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

T= sqrt[s(s−a)(s−b)(s−c)]
​T= sqrt[8.5(8.5−6)(8.5−6)(8.5−5)]
​T= 185.94===13.64

Area: T = 13.636 u^2
Perimeter: p = 17
Semiperimeter: s = 8.5

The 3 positive consecutive integers are: 567

[5 x 7] ==[3 x 6 + 17] ==35 == 35

The correct answer is 5.

The sum of all the numbers is 1053488.

Sum(1, 4, 2, 8, 3, 12, 4, 16, 5, 20, 6, 24, 7, 28, 8, 32, 9, 36, 10, 40, 11, 44, 12, 48, 13, 52, 14, 56, 15, 60, 16, 64, 17, 68, 18, 72, 19, 76, 20, 80, 21, 84, 22, 88, 23, 92, 24, 96, 25, 100)==1,625

The ODD terms are: 1, 2, 3, 4, 5,........25

The EVEN terms are: 4, 8, 12, 16,.......100 [Or multiples of 4]

(- 1) x 2020/2 + 2021 ==(- 1) x 1010 + 2021 ==- 1010 + 2021 ==1011

The sum is 1800.

The sum is 1015.

??? What ???

Thanks!

If 16 centimeters of wire costs 48 cents, then we can simplify the ratio to 1 centimeter of wire costs 3 cents. Then, we multiply 1 by 11, and 3 by 11, to get 11 centimeters of wire costs 33 cents.

10 - 99 : 4
100 - 999 : 69
1000 - 9999 : 603
10000 - 99999 : 5817
100000 - 999999 : 60378
1000000 - 9999999 : 602133
10000000 - 99999999 : 5987904
100000000 - 999999999 : 59994303
1000000000 - 9999999999 : 600225342

It is the last sum that you want ==600,225,342

It is not possible for a function to have different values for the same input. f(3) cannot be 0 and 28 at the same time.

Plotting it in a graphical calculator gives the range as \((-\infty, +\infty)\).

If n/21 is reducible, then either n is a multiple of 3 or n is a multiple of 7.

There are floor(100/3) = 33 multiples of 3 from 1 to 100, and there are floor(100/7) = 14 multiples of 7 from 1 to 100.

But we have double counted floor(100/21) = 4 multiples of 21, so the total number of integers n with 1 <= n <= 100 such that n/21 is reducible is 33 + 14 - 4 = 43.