### Simplifying fraction with nested radicals and fractions

This is my first question here and on a stack exchange in general. I hope my question is precise enough. I have spent a good 15min searching the forum but didn't manage to understand the below.I am confused by how the fraction to the left of the equal sign is simplified to $\frac{\sqrt 2}{2}$.$\frac{\sqrt \frac{\frac{a}{2}}{\sqrt 2}}{\sqrt \frac{a}{\sqrt 2}}=\frac{\sqrt 2}{2}$What steps can I take to get from $\frac{\sqrt \frac{\frac{a}{2}}{\sqrt 2}}{\sqrt \frac{a}{\sqrt 2}}$ to $\frac{\sqrt 2}{2}$ ? Thanks in advance for your help...Read more

### Simplifiable Sum of two Unsimplfiable Fractions

I am programing something that is supposed to represent fractions. I will try to leave out as much of the programming specifics as I can because my problem is a math problem not a programming problem. First off I will explain that numbers held in memory have maximum and minimum values meaning they are finite. The maximum value is defined as any number that can be expressed as max = (2^b) - 1where 'b' is a positive integer. And the minimum value will be defined as the opposite of the maximum value. So that the absolute value of the minimum value...Read more

### fractions - How do you calculate/prove the length of n, the number of non-repeating digits preceeding a periodic sequence of a fractional repeating decimal

Is there a way to calculate the number of non-repeating digits that precede the periodic repeating portion of a decimal expansion? For example:1/6 = 0.1666.... (there is 1 non repeating digit) **(Correction)1/12 = 0.08333... (there are 2 non repeating digits)7/12 = 0.58333....(there are 2 non repeating digits)1/96 = 0.01041666..(there are 5 non repeating digits)Do any forumulas exist for predicting the maximum length n, of the number of non repeating digits preceding the repeating portion? I know that if the denominator of a fraction is n, t...Read more

### Express $x=0.\overline{31}_5$ as a fraction in lowest terms

Given $x=0.\overline{31}_5$, find the value of $x$, expressed as a fraction in lowest terms.I tried to change it into base $10$, but I don't think it's possible with fractions. So please help I'll appreciate it. Also I'm in 7th grade (easy solutions please) and no copying other people's answer (I've seen it in other problems)....Read more

### fractions - $\frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y}$

If for three distinct positive numbers $x$, $y$, and $z$, $$\frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y}$$then find the value of $\frac{x}{y}$I have tried all types of manipulations, even quadratics but can seem to get the answer. Please help!! BTW im a 7th grader so easy solutions will be appreciated....Read more

### fractions - Simplify a rational identity

Simplify: $$\frac{\dfrac{a}{b}-\dfrac{b}{a}}{1+\dfrac{b}{a}}$$I have a feeling the solution has to do with factoring, but I'm really not sure, and would appreciate any help....Read more

### rational functions - Simplify the algebraic fraction

How to simplify the algebraic fraction$\frac{2}{x+5} + \frac{3}{x-2} + \frac{1}{x}$I get the answer as $\frac{2(3x^2 +7x-5)}{x(x+5)(x-2)}$Is this right? Or how can I simplify this further?...Read more

### Simplify the algebraic fractions

How could I simplify $\dfrac 8{x+5} - \dfrac 3{5-x}$? I know we should find the LCM for the denominators and then simplify, but this question is little confusing....Read more

### fractions - The product of two repeating decimals

In An exceptional talent for calculative thinking, (IML Hunter, 1962), Professor Aitken explained how he found the decimal expansion of 1/851: 851 is 23 times 37. I use this fact as follows. 1/37 is 0.027027027027. . .. This I divide mentally by 23... 1/37 recurs at three places, 1/23 recurs at twenty-two places, the lowest common multiple of 3 and 22 is 66, whence I know that there is a recurring period of 66 places.’I thought this is a general fact: two rationals with decimal periods a and b must have a decimal period of ab, but there is a c...Read more

### How to solve this algebraic fraction?

Could someone please help me with solving this algebraic fraction. I tried it a few times and I got the wrong answer all of the times. My brother also tried, who had recently finished Matric and he is very good at Maths. He also got the wrong answer. I'm completely lost with this one!algebraic fractionIf you can't read the image (I apologize, I can't write neatly on paint on a computer!), here is a "worded fraction":$$\frac{a^2 + ab}{2b - a}: \frac{a}{2a - 4b}$$The answer we got from our teacher was $2(a - 2b)$I'm sorry if this isn't clear enou...Read more

### Why are fractions "multiplied across"?

This question already has an answer here: Multiplying and adding fractions 1 answer Understanding the multiplication of fractions [duplicate] 2 answers...Read more

### Why does cross-cancellation of fractions in multiplication work?

I'm reviewing my arithmetic and right now I'm at fractions, I'm just having a little bit of problem "visualizing" why cancellation works in multiplying fractions, I know how it works, it's just the why. Take for example.$$\frac{2}{3}\cdot\frac{3}{4}=\frac{1}{3}\cdot\frac{3}{2}=\frac{1\cdot1}{1\cdot2}=\frac{1}{2}.$$ Through cancellation we know this ends in 1/2 because the common factor of the numerator "2" and denomaninator "4" is 2, so we divide both by "2" and end with new numbers in place, we also know that the common factor of numerator "3"...Read more

### How to calculate a repeating decimal for any fraction?

I have been struggling for a while to try to code a program to convert any fraction 1/n to a repeating decimal. So far, my program works only for numbers that end in 1, 3, 7, or 9 (n cannot divide 2 or 5, since those numbers divide 10, our numeral base). Here is my program:click hereWhat the program does is it finds a long string of 9s that is divisible by n. So for 1/7, you keep looking through strings of 9s until you get 999999, which is divisible by 7. When you divide that by 7, you get 142857, so thus, 1/7 is equal to 0.142857 with all digi...Read more

### Simplify fraction in numerator when fraction is part of sum

I have the result of a quotient rule:$$\frac{\frac{x}{8+x}-ln8}{x^2}$$Should I just leave it? Would it be appropriate to separate the fraction into a product of fractions with denominator $x^2$, simplify the left side, and give the result as a product of two fractions?Any soln using addition or subtraction of fractions is unacceptable and messier than just leaving as is....Read more

### fractions - How do I solve this rational expression?

So the expression is:$$\frac1{2b} + \frac b2$$Apparently the answer is $\frac{1 + b^2}{ 2b}$.I came up with $2b$ by multiplying the first fraction by $2$ and the second fraction by $2b.$I got:$$\frac{2+ 2b^2}{2(2b)}$$ which simplifies to $\frac{2b^2}{2b}$ which is where I got $2b$ from. Can someone help me with this problem and the LCD?...Read more