Tagged Questions
4
votes
1answer
58 views
Simplifying $y=2^{2/3} + 2^{-1/3}$
I am working on a calculus problem where I have to find the local minimum. The value I got was $$y=2^{2/3} + 2^{-1/3}.$$ I simplified it and got this:
$$ y=2^{2/3} + \frac{1}{2^{1/3}}$$
...
0
votes
1answer
35 views
Finding that probability of the event is small
Let $x_1, \ldots, x_n$ be Bernoulli random variables with the probability of success $P(x_i=1)=p$.
Let $\epsilon>0$.
Show that probability
$$
P\left(\left|\sum_{i=1}^nx_i-p\right|> ...
3
votes
4answers
70 views
Proof of $\sqrt{2^{2^k}} = 2^{2^{k-1}}$?
It's quite easy to observe that for $k \ge 0$:
$$
\begin{align}
2^{2^k} &= 4, 16, 256, 65536, \dots\\
\sqrt{2^{2^k}} &= 2, 4, 16, 256,\dots
\end{align}
$$
More in general:
$$
\sqrt{2^{2^k}} ...
9
votes
2answers
257 views
1
vote
2answers
65 views
Simplifying negative exponents when there are multiple terms.
The problem is to simplify
$$\left(\dfrac{1+3z}{3z}\right)$$
I know that when I have $$\dfrac{1}{x}$$ I can bring the $x$ from the denominator to the numerator by changing it to a negative power. How ...
6
votes
3answers
168 views
How do you compute negative numbers to fractional powers?
My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure ...
0
votes
2answers
43 views
On exponentials of matrices
How can I prove that $|e^{A}|\leq e^{|A|}$? I guess i'm having trouble with the definition!
A is a square (complex or real) matrix.
1
vote
2answers
92 views
$\det(\exp X)=e^{\mathrm{Tr}\, X}$ for 2 dimensional matrices
I want to prove that for $X\in M_2(\mathbb{R})$ the formula $\det(\exp X)=e^{\mathrm{Tr}\, X}$ holds, writing $X$ in normal form gives $X=PJP^{-1}$, where $J$ is the Jordan matrix, now $\exp ...
0
votes
3answers
99 views
Manipulating Exponents
I'm doing my homework and there are a couple of things that I am having trouble grasping. All my homework asks is that I simplify the exponents. For example: ...
1
vote
2answers
111 views
Simplifying $\frac{2^{n + 4} + 2^{n + 2} + 2^{n - 1}}{2^{n - 2} + 2^{n - 1}}$
I'm stuck in the follow equation:
$$\dfrac{2^{n + 4} + 2^{n + 2} + 2^{n - 1}}{2^{n - 2} + 2^{n - 1}}$$
As all the bases are equal, I got $\dfrac{3n + 5}{2n - 3}$
Where I've to go now ?
Thanks
...
0
votes
2answers
225 views
Graphing Fractional Exponents
$f(x)=x^\frac{5}{3}-5x^\frac{2}{3}$
is the same as :
$f(x)=(\sqrt[3]x)^5-(\sqrt[3]{5x})^2$
Except, with the first equation, my calculator returns an error for negative values of $x$ (We are ...
0
votes
2answers
147 views
Simplifying exponents, multiplication, and addition
How can you get $10^{n+1}$ from $9\cdot 10^n+10^n$? This is part of a proof I am working on.
4
votes
3answers
148 views
Prove $\frac{(5^{x-1}+5^{x+1})^2}{25^{x-1}+25^{x+1}}=\frac{338}{313}$
Q. Prove
$$\frac{(5^{x-1}+5^{x+1})^2}{25^{x-1}+25^{x+1}}=\frac{338}{313}$$
My try: expand and got:
$$\frac{5^{2x-2}+2(5^{x^2-1})+5^{2x+2}}{5^{2x-2}+5^{2x+2}}$$
Now what? I find my pre-calculus ...
0
votes
1answer
52 views
Simplifying with negative exponents $(-11a^2)(-4a^{-7})$
$$(-11a^2)(-4a^{-7})$$
Can someone reformat, $a$ is second set of parenthesis is to the $-7$ power.
Change to reciprocal so we get
$$\left(\frac{1}{-4a}\right)^7 * \frac{11a}{1} = $$ confused
...
0
votes
1answer
54 views
Re-writing a logarithm to a power
Given:
$$(4\ln x)^2$$
Is this simplified to $8\ln x$, (multiplying the expression by 2),
$32\ln x$, (square $4$ ($16$), then $\ln x$ ($2\ln x$) and combine again),
or something else?
Just to be ...
1
vote
2answers
311 views
