Master Fraction Math Problems with Expert Help

Adding fractions with exponents
$3^{5} + \frac{1}{3^{5}} = ?$
My first instinct was to rewrite the second term as $3^{- 5}$ . Since the base is $3$ , rewrite as $3^{5 + - 5}$ . It simplifies to $3^{0} = 1$ . Apparently this is incorrect. Can anyone please explain why?

misurrosne 2022-06-13

Simplifying square numbers in a fraction (elementary school lol)
I have the following question in a practice booklet:
If $\frac{2^{26} - 2^{23}}{2^{26} + 2^{23}}$ = $\frac{x}{9}$ What is $x$ ?
I know the answer is 7 because it's easy enough when you manually figure out $2^{26}$ etc. but is there a faster way? Can I simplify this down somehow? .

Jase Howe 2022-06-13

I'm comfortable with fractions like $\frac{- 3}{8}$ being the same as $\frac{3}{- 8}$ (though I'd think the latter anachronistic and would in any case probably prefer to write either of those two as $- \frac{3}{8}$ ), and of course I'm comfortable with improper fractions like $\frac{- 8}{3}$ being the mixed number $- 2 \frac{2}{3}$
However, if I'm trying to teach a computer how to handle negative vulgar fractions, I should also consider the remaining cases, so how should I interpret:
1. $2 \frac{- 2}{3}$
2. $2 \frac{2}{- 3}$
3. $2 \frac{- 2}{- 3}$
4. $- 2 \frac{- 2}{3}$
5. $- 2 \frac{2}{- 3}$
6. $- 2 \frac{- 2}{- 3}$
I'm considering the logical approach, by inference from $2 \frac{2}{3}$ and $- 2 \frac{2}{3}$ so I'd get:
1. $1 \frac{1}{3}$
2. $1 \frac{1}{3}$
3. $1 \frac{2}{3}$
4. $- 1 \frac{1}{3}$
5. $- 1 \frac{1}{3}$
6. $- 2 \frac{2}{3}$

Brunton39 2022-06-13

Help with simplification of a rational expression (with fractional powers)
Can you please help me see what I don't see yet. Here's a problem from a high school textbook (ISBN 978-5-488-02046-7 p.9, #1.029):
$\frac{(a^{1 / m} - a^{1 / n})^{2} \cdot 4 a^{(m + n) / m n}}{(a^{2 / m} - a^{2 / n}) (\sqrt[m]{a^{m + 1}} + \sqrt[n]{a^{n + 1}})}$
Here's my try at it:
$\frac{(a^{1 / m} - a^{1 / n}) (a^{1 / m} - a^{1 / n}) \cdot 4 a^{(1 / m) + (1 / n)}}{(a^{1 / m} - a^{1 / n}) (a^{1 / m} + a^{1 / n}) \cdot a (a^{1 / m} + a^{1 / n})}$
...which is
$\frac{(a^{1 / m} - a^{1 / n}) \cdot 4 a^{(1 / m) + (1 / n)}}{a (a^{1 / m} + a^{1 / n})^{2}}$
Wolfram Alpha's simplify stops here, too. I don't see where to go from here. The final form, according to the book, is this:
$\frac{1}{a (a^{1 / m} - a^{1 / n})}$
How did they do it?

Emmy Knox 2022-06-13

Calculating pOH
If [OH-]=10^-pOH and [OH-]= 0.003 then what does pOH equal? I know this is simple but I just can't figure out how to do this calculation. Any help would be appreciated.

watch5826c 2022-06-13

How to calculate an $n$ digit decimal approximation of a fraction?
Say I have a number $x \in Q$ which possibly cannot be writen exactly in decimal form, but at least $x \notin Z$ ( $x$ is not an integer).
How do I calculate the first $n$ digits of this number $x$ , for a given $n$ ? Also, how do I know if it has a finite or infinite amount of (nonzero) digits in decimal form?
For example, if $x = \frac{1000}{1001}$ , how do I calculate the first $n = 10$ digits of this number?
I know this question sounds really basic, but since it is basically always done by means of a calculator or other computing device, I've never really done it before.

Jeffery Clements 2022-06-12

Partial Fractions Question: $\frac{1}{s (n - s)}$
How do you decompose the fraction to two fractions?
(s and n are variables).

Kassandra Ross 2022-06-12

How exactly is this happening?
I was studying Derivative and my book says if:
$y = \sqrt{(1 + x^{2})^{3}}$
Then its derivative is:
$y^{'} = \frac{3 (2 x) (1 + x^{2})^{2}}{2 \sqrt{(1 + x^{2})^{- 1}}} = 3 x \sqrt{(1 + x^{2})}$
I can't understand how the writer has changed the first derivative fraction into the second one. In other words, how did he simplify?
Note: I'm really really basic, so please explain in details.
Thank you.

Llubanipo 2022-06-12

How to do 24.89 / 26.15 with long division?
How to do 24.89 / 26.15 with long division? Question is to find applied tax if 24.89 is pre-tax and 26.15 is post-tax. I'm aware of how to solve this with a calculator (1-(24.89/26.15))= ~.0482 = ~ 4-5 cents

gvaldytist 2022-06-11

show that $\frac{c^{2}}{a} + \frac{a^{2}}{c} + 16 \sqrt{a c} \geq 9 (a + c)$
Let $a, c > 0$ show that
$\frac{c^{2}}{a} + \frac{a^{2}}{c} + 16 \sqrt{a c} \geq 9 (a + c)$
It seem AM-GM inequality.How?

Dale Tate 2022-06-11

How to solve advanced fraction problem with algebra
Good evening guys, ok so ive been stumped on this for some time and am a bit rusty with my fractions so i was wondering if i could get some help on answering this
I have to try and write this fraction on its own and in its most basic form (simple)
The problem:
$\frac{(\frac{a^{3} - 3 a^{2}}{8 a^{2} - 4 a})}{(\frac{a^{2} - 9}{2 a^{2} + 5 a - 3})}$
Am really stuck, any help would be appreciated!

Emmy Knox 2022-06-11

Proving functions to be Big Oh
How do I determine if there exists a function $f$ , such that
$f (n) = O (\log n),$
but
$2^{f (n)} \neq O (n) .$
Is true or false?
I tried using the c and No method but cannot come up with a solution

Brunton39 2022-06-10

Convergence of a series
Does this series converge?
$\sum_{x = 2}^{n} {(\frac{1}{x})}^{(\frac{1}{x})}$
I tried hard but stil had problems...
Could someone help me?

taghdh9 2022-06-10

Solving a rational equation with multiple and nested fractions
This is the equation to solve: $\frac{\frac{x + \frac{1}{2}}{\frac{1}{2} + \frac{x}{3}}}{\frac{1}{4} + \frac{x}{5}} = 3$
What I did:
$x + \frac{1}{2} = \frac{2 x + 1}{2}$
$\frac{x}{3} + \frac{1}{2} = \frac{2 x + 3}{6}$
$\frac{x}{5} + \frac{1}{4} = \frac{4 x + 5}{20}$
$\frac{2 x + 1}{2} \div \frac{2 x + 3}{6} = \frac{2 x + 1}{2} \times \frac{6}{2 x + 3} = \frac{6 x + 3}{2 x + 3}$
$\frac{6 x + 3}{2 x + 3} \div \frac{4 x + 5}{20} = \frac{6 x + 3}{2 x + 3} \times \frac{20}{4 x + 5} = 3$
$⟹ \frac{120 x + 60}{(2 x + 3) (4 x + 5)} = 3$
I know how to solve the equation. But right now I tried several times and I got wrong answers.
So I appreciate your help. Thank you.

Yahir Tucker 2022-06-10

Find the smallest number $α$ , such that for all $x, y, z$
$α (x^{2} - x + 1) (y^{2} - y + 1) (z^{2} - z + 1) \geq (x y z)^{2} - x y z + 1$
My work so far:
Let $f (t) = t^{2} - t + 1$ . Then $f (t) \geq \frac{3}{4}$
If $x = 0, y = z = \frac{1}{2}$ , then
$α \geq \frac{16}{9}$

Reed Eaton 2022-06-10

Probability of 2 students being chosen the both have under 100 books at home
Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home?
Data provided is the amount of books each student has at home. There are 6 out of the 15 students that have less than 100 books:
33, 47, 48, 52, 56, 66, 173, 186, 251, 300, 323, 417, 450, 1280, 2000
I tried doing 6/15 x 5/14 = 30/210 or a 14% chance. Is this correct?
Thank you!

polivijuye 2022-06-09

Trending proof for fairly simple fraction
(Please humour the physicist!)
Why does $(\frac{1 - a}{1 - a^{b}}) \to \frac{1}{b}$ as $a \to 1$ ?
This came from a calculation involving flow measurement of gases, and although I can see and compute the answer, I don't understand how this general result occurs.

Jasmin Pineda 2022-06-08

Why is $\frac{\sqrt{x + 1} - 1}{x}$ equal to $\frac{1}{\sqrt{x + 1} + 1}$ ?
I'm working with the expression
$\frac{\sqrt{x + 1} - 1}{x} .$
According to Wolfram Alpha "alternate form" section (http://www.wolframalpha.com/input/?i=%28%28x%2B1%29%5E1%2F2-1%29%2Fx) it is equal to
$\frac{1}{\sqrt{x + 1} + 1} .$
How can I go from one expression to the other?

migongoniwt 2022-06-08

Square root fraction confusion
I was doing math for school and got to something that really confused me. With having the rule $\frac{2}{4} = \frac{4}{8}$ (or some simular fraction equation) in mind, I got to the following confusing equation:
$\frac{\sqrt{3}}{3} = \frac{(\sqrt{3})^{2}}{3^{2}} = \frac{3}{9} = \frac{1}{3}$
This is really confusing me since $\frac{\sqrt{3}}{3}$ can never equal $\frac{1}{3}$ . Where am I going wrong?

Jayla Christensen 2022-06-08

Prove that $\frac{\frac{1}{11} + \frac{1}{12} + \dots + \frac{1}{200}}{\frac{1}{10 * 11} + \frac{1}{11 * 12} + \dots + \frac{1}{19 * 20}} > 19$
My attempt:The denominator is $\frac{1}{20}$ using telescopic series.
$\frac{1}{10 * 11} + \frac{1}{11 * 12} + \dots + \frac{1}{19 * 20} = \frac{1}{10} - \frac{1}{11} + \frac{1}{11} - \frac{1}{12} + \dots - \frac{1}{20} = \frac{1}{10} - \frac{1}{20} = \frac{1}{20}$
I think we could Prove that the top part of fraction is bigger than $1$ but I don't know how.

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