MaxWong

\((a + b)^2 = (a - b)^2\\ (a + b)^2 - (a - b)^2 = 0\\ \text{Using }a^2 - b^2 = (a - b)(a + b),\\ (2a)(2b) = 0\\ ab = 0\\ a = 0\text{ or } b = 0\)

AA9_{12} is right.

The WolframAlpha computation: https://www.wolframalpha.com/input/?i=14A6+base+12+-+5B9+base+12

Assuming you meant \(y = \dfrac25 x\),

Substituting (p, q) into the equation, \(q = \dfrac{2p}5\),

Because pq = 90, p(2p/5) = 90, which means p² = 225

Taking square roots on both sides, p = 15.

This has been posted few days ago.

https://web2.0calc.com/questions/pls-help-due-today_1

We can rewrite this as 

\(\begin{cases}\log_w m = \dfrac1{12}\\\log_w n = \dfrac1{16}\\\log_w p = \dfrac1{36}\\\log_{w} mnpq = \dfrac1{72}\end{cases}\)

By log properties, 

\(\log_w m + \log_w n + \log_w p + \log_w q = \dfrac1{72}\\ \dfrac1{12} + \dfrac1{16} + \dfrac1{36} + \dfrac1{\log_q w} = \dfrac1{72}\)

The rest is trivial.

Ans: \(\log_q w = -\dfrac{144}{23}\)

Hint: Consider the ratio of side lengths of each triangle.

\(f(x) = (x^2 + 6x + 9)^{50} - 4x + 3 = (x + 3)^{100} - 4x + 3\)

Substituting \(r_1, r_2,\cdots,r_{100}\) into f(x) = 0, we have

\(\begin{cases}\left(r_1 + 3\right)^{100} - 4r_1 + 3 = 0\\\left(r_2 + 3\right)^{100} - 4r_2 + 3 = 0\\\cdots\\\left(r_{100} + 3\right)^{100} - 4r_{100} + 3 = 0\end{cases}\)

Moving all the terms to the right hand side and adding them up,

\((r_1 + 3)^{100} + (r_2 + 3)^{100} + \cdots + (r_{100} + 3)^{100} = 4(r_1 + r_2 + \cdots + r_{100}) - 300\)

That means the required answer is 4(sum of roots) - 300.

By Vieta's formula, \(\text{sum of roots} = -\dfrac{\text{coefficient of }x^{99}}{\text{coefficient of }x^{100}}\).

Using binomial theorem, the coefficient of x¹⁰⁰ is 1 and the coefficient of x⁹⁹ is 300.

Therefore \(r_1 + r_2 + \cdots + r_{100} = -300\).

Therefore \((r_1 + 3)^{100} + (r_2 + 3)^{100} + \cdots + (r_{100} + 3)^{100} = 4(-300) - 300 = \boxed{-1500}\)

Let t = 1/x.

\(25^{-2} = \dfrac{5^{48t}}{5^{26t} \cdot 25^{17t}}\)

Rewrite 25^n into 5^(2n),

\(5^{-4} = \dfrac{5^{48t}}{5^{26t}\cdot 5^{34t}}\)

By law of indices,

\(5^{-4}= 5^{-12t}\)

Equating the powers would give t = 1/3, which means x = 3.

The bag of loonies are worth $400 in total ⇔ There are 400 loonies in the bag.

But one loonie = 4 dimes in terms of mass, so if the bag of dimes has the same mass as a bag of loonies, then there should be (4)(400) dimes, which is 1600 dimes.

The bag of dimes is worth $1600 (0.10) = $160.