# Tag Info

### Graph Transformation

$y = f(x+c)$ translates the graph by $-c$ in the x direction. $y = f(x) + c$ translates the graph by $c$ in the y direction. $y = f(cx)$ stretches the graph by a factor of $\frac{1}{c}$ in the x ...

### Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

This answer uses the same sort of reasoning as given in the answer by zyx. I show that we can get a bit more out of the recursion relation satisfied by the function $S(n)$ and their generalizations ...
1 vote
Accepted

### Evaluate ${\cos ^{ - 1}}\sqrt {1 - {x^2}} =$ for $x<0$

Because $x=\sin \theta <0$ and $\cos \theta = \sqrt{1-x^2} >0$, angle $\theta$ is in the fourth quadrant. Thus, $\cos^{-1} |\cos \theta|=-\theta$

### Solve the equation $6(3^{2x})=2^{(x+1)}$

Another way to approach it for the sake of curiosity: \begin{align*} 6\cdot3^{2x} = 2^{x+1} & \Longleftrightarrow 3\cdot 9^{x} = 2^{x}\\\\ & \Longleftrightarrow \left(\frac{9}{2}\right)^{x} = \...

### Is there a variant of the dot-product operation that returns $\frac{a_1}{b_1} + \frac{a_2}{b_2}$ from vectors $[a_1,a_2]$ and $[b_1, b_2]$?

To answer the asked question: NO. While there are many inverse simplification formulas that exist, none can be resolved to fulfill your requested outcome for all input values. I agree that the ...
1 vote

### Solve the equation $6(3^{2x})=2^{(x+1)}$

The answer to your first question is yes: $$2x \ln 3 = x \ln 3^2 = x \ln 9.$$ The answer to your second question is no: $$x \ln 2 + \ln 2 \ne x \ln 4,$$ because if this were true, it would ...
1 vote

### Find the sum of radicals without squaring ?Is that impossible?

In this answer I tried to generalize the answer given above. Generalization: Let, $$\sqrt {a\pm\sqrt b}=\sqrt m\pm\sqrt n,\,\,\,m\ge n$$ and $$A=\sqrt {a+\sqrt b}+\sqrt {a-\sqrt b}=2\sqrt m$$ Then we ...