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Least common denominator:
((a*sqrt(a)+b*sqrt(b))/(sqrt(a)+sqrt(b)))/(a-b)+(2*sqrt(b))/(sqrt(a)+sqrt(b))
sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))
(n/((p-4)*(p-4)))-(8/((4-p)*(4-p)))
m-4/m-m-3/m+1
Factor polynomial:
x^4-3*x
x^4/2-x-x^2/2
x^4-2*x-6*x^2+5*x+2
x^4-2*x+4*x^2
Factor squared:
x^2-x*t+t^2
x^2-x*a+12*a^2
-x^2+x+8
-x^2-x*b+15*b^2
How do you in partial fractions?:
1/(x^4+x^2)
x^2/(1-x^4)
(w*i*0.00005)/(1+w*i*0.00005)
w^2/(w-6)-12*w-36/(w-6)
How do you in partial fractions?:
sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))
Identical expressions
sqrt(b^ two *((one -(b^ two -a^ two)/(b^ two +a^ two))^ two +(two *a-b*sqrt(one -((b-a)*(b+a)/(b^ two +a^ two))^ two)^ two)))
square root of (b squared multiply by ((1 minus (b squared minus a squared ) divide by (b squared plus a squared )) squared plus (2 multiply by a minus b multiply by square root of (1 minus ((b minus a) multiply by (b plus a) divide by (b squared plus a squared )) squared ) squared )))
square root of (b to the power of two multiply by ((one minus (b to the power of two minus a to the power of two) divide by (b to the power of two plus a to the power of two)) to the power of two plus (two multiply by a minus b multiply by square root of (one minus ((b minus a) multiply by (b plus a) divide by (b to the power of two plus a to the power of two)) to the power of two) to the power of two)))
√(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a-b*√(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))
sqrt(b2*((1-(b2-a2)/(b2+a2))2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b2+a2))2)2)))
sqrtb2*1-b2-a2/b2+a22+2*a-b*sqrt1-b-a*b+a/b2+a222
sqrt(b²*((1-(b²-a²)/(b²+a²))²+(2*a-b*sqrt(1-((b-a)*(b+a)/(b²+a²))²)²)))
sqrt(b to the power of 2*((1-(b to the power of 2-a to the power of 2)/(b to the power of 2+a to the power of 2)) to the power of 2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b to the power of 2+a to the power of 2)) to the power of 2) to the power of 2)))
sqrt(b^2((1-(b^2-a^2)/(b^2+a^2))^2+(2a-bsqrt(1-((b-a)(b+a)/(b^2+a^2))^2)^2)))
sqrt(b2((1-(b2-a2)/(b2+a2))2+(2a-bsqrt(1-((b-a)(b+a)/(b2+a2))2)2)))
sqrtb21-b2-a2/b2+a22+2a-bsqrt1-b-ab+a/b2+a222
sqrtb^21-b^2-a^2/b^2+a^2^2+2a-bsqrt1-b-ab+a/b^2+a^2^2^2
sqrt(b^2*((1-(b^2-a^2) divide by (b^2+a^2))^2+(2*a-b*sqrt(1-((b-a)*(b+a) divide by (b^2+a^2))^2)^2)))
Similar expressions
sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2-(2*a-b*sqrt(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))
sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1-((b-a)*(b-a)/(b^2+a^2))^2)^2)))
sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1-((b+a)*(b+a)/(b^2+a^2))^2)^2)))
sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a+b*sqrt(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))
sqrt(b^2*((1-(b^2+a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))
sqrt(b^2*((1+(b^2-a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))
sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b^2-a^2))^2)^2)))
sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1+((b-a)*(b+a)/(b^2+a^2))^2)^2)))
sqrt(b^2*((1-(b^2-a^2)/(b^2-a^2))^2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))

Expression simplification/ Least common denominator/ sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))

Least common denominator sqrt(b^2((1-(b^2-a^2)/(b^2+a^2))^2+(2a-bsqrt(1-((b-a)(b+a)/(b^2+a^2))^2)^2)))

The solution

You have entered [src]

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       /    /                                                       2\ 
      /     |             2                 ________________________ | 
     /      |/     2    2\                 /                      2  | 
    /     2 ||    b  - a |                /      /(b - a)*(b + a)\   | 
   /     b *||1 - -------|  + 2*a - b*   /   1 - |---------------|   | 
  /         ||     2    2|              /        |     2    2    |   | 
\/          \\    b  + a /            \/         \    b  + a     /   /

$$\sqrt{b^{2} \left(\left(2 a - b \left(\sqrt{1 - \left(\frac{\left(- a + b\right) \left(a + b\right)}{a^{2} + b^{2}}\right)^{2}}\right)^{2}\right) + \left(- \frac{- a^{2} + b^{2}}{a^{2} + b^{2}} + 1\right)^{2}\right)}$$

sqrt(b^2*((1 - (b^2 - a^2)/(b^2 + a^2))^2 + 2*a - b*(sqrt(1 - (((b - a)*(b + a))/(b^2 + a^2))^2))^2))

General simplification [src]

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       /    /         /           2                    \                2\ 
      /   2 |   4     |  / 2    2\           2        2|       / 2    2\ | 
     /   b *\4*a  + b*\- \a  + b /  + (a + b) *(b - a) / + 2*a*\a  + b / / 
    /    ----------------------------------------------------------------- 
   /                                          2                            
  /                                  / 2    2\                             
\/                                   \a  + b /

$$\sqrt{\frac{b^{2} \left(4 a^{4} + 2 a \left(a^{2} + b^{2}\right)^{2} + b \left(\left(- a + b\right)^{2} \left(a + b\right)^{2} - \left(a^{2} + b^{2}\right)^{2}\right)\right)}{\left(a^{2} + b^{2}\right)^{2}}}$$

sqrt(b^2*(4*a^4 + b*(-(a^2 + b^2)^2 + (a + b)^2*(b - a)^2) + 2*a*(a^2 + b^2)^2)/(a^2 + b^2)^2)

Fraction decomposition [src]

sqrt(b^2 - b^3 + b^6/(a^4 + b^4 + 2*a^2*b^2) + b^7/(a^4 + b^4 + 2*a^2*b^2) - 2*b^4/(b^2 + a^2) + 2*a*b^2 + a^4*b^2/(a^4 + b^4 + 2*a^2*b^2) + a^4*b^3/(a^4 + b^4 + 2*a^2*b^2) - 2*a^2*b^4/(a^4 + b^4 + 2*a^2*b^2) - 2*a^2*b^5/(a^4 + b^4 + 2*a^2*b^2) + 2*a^2*b^2/(b^2 + a^2))

$$\sqrt{\frac{a^{4} b^{3}}{a^{4} + 2 a^{2} b^{2} + b^{4}} + \frac{a^{4} b^{2}}{a^{4} + 2 a^{2} b^{2} + b^{4}} - \frac{2 a^{2} b^{5}}{a^{4} + 2 a^{2} b^{2} + b^{4}} - \frac{2 a^{2} b^{4}}{a^{4} + 2 a^{2} b^{2} + b^{4}} + \frac{2 a^{2} b^{2}}{a^{2} + b^{2}} + 2 a b^{2} + \frac{b^{7}}{a^{4} + 2 a^{2} b^{2} + b^{4}} + \frac{b^{6}}{a^{4} + 2 a^{2} b^{2} + b^{4}} - \frac{2 b^{4}}{a^{2} + b^{2}} - b^{3} + b^{2}}$$

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     /                    6                   7               4                    4  2                4  3                 2  4                2  5           2  2 
    /   2    3           b                   b             2*b          2         a *b                a *b               2*a *b              2*a *b         2*a *b  
   /   b  - b  + ----------------- + ----------------- - ------- + 2*a*b  + ----------------- + ----------------- - ----------------- - ----------------- + ------- 
  /               4    4      2  2    4    4      2  2    2    2             4    4      2  2    4    4      2  2    4    4      2  2    4    4      2  2    2    2 
\/               a  + b  + 2*a *b    a  + b  + 2*a *b    b  + a             a  + b  + 2*a *b    a  + b  + 2*a *b    a  + b  + 2*a *b    a  + b  + 2*a *b    b  + a

Numerical answer [src]

(b^2*((1.0 - (b^2 - a^2)/(a^2 + b^2))^2 + 2.0*a - b*(1.0 - (a + b)^2*(b - a)^2/(a^2 + b^2)^2)^1.0))^0.5

(b^2*((1.0 - (b^2 - a^2)/(a^2 + b^2))^2 + 2.0*a - b*(1.0 - (a + b)^2*(b - a)^2/(a^2 + b^2)^2)^1.0))^0.5

Expand expression [src]

                __________________________________________________
               /              2                                   
   ____       /  /     2    2\            /           2        2\ 
  /  2       /   |    b  - a |            |    (b + a) *(b - a) | 
\/  b  *    /    |1 - -------|  + 2*a - b*|1 - -----------------| 
           /     |     2    2|            |                 2   | 
          /      \    b  + a /            |        / 2    2\    | 
        \/                                \        \b  + a /    /

$$\sqrt{2 a - b \left(- \frac{\left(- a + b\right)^{2} \left(a + b\right)^{2}}{\left(a^{2} + b^{2}\right)^{2}} + 1\right) + \left(- \frac{- a^{2} + b^{2}}{a^{2} + b^{2}} + 1\right)^{2}} \sqrt{b^{2}}$$

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       /    /             2                                  \ 
      /     |/     2    2\            /           2        2\| 
     /    2 ||    b  - a |            |    (b + a) *(b - a) || 
    /    b *||1 - -------|  + 2*a - b*|1 - -----------------|| 
   /        ||     2    2|            |                 2   || 
  /         |\    b  + a /            |        / 2    2\    || 
\/          \                         \        \b  + a /    //

$$\sqrt{b^{2} \left(2 a - b \left(- \frac{\left(- a + b\right)^{2} \left(a + b\right)^{2}}{\left(a^{2} + b^{2}\right)^{2}} + 1\right) + \left(- \frac{- a^{2} + b^{2}}{a^{2} + b^{2}} + 1\right)^{2}\right)}$$

sqrt(b^2*((1 - (b^2 - a^2)/(b^2 + a^2))^2 + 2*a - b*(1 - (b + a)^2*(b - a)^2/(b^2 + a^2)^2)))

Assemble expression [src]

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       /    /             2                                   \ 
      /     |/     2    2\            /            2        2\| 
     /    2 ||    b  - a |            |     (a + b) *(b - a) || 
    /    b *||1 - -------|  + 2*a + b*|-1 + -----------------|| 
   /        ||     2    2|            |                  2   || 
  /         |\    a  + b /            |         / 2    2\    || 
\/          \                         \         \a  + b /    //

$$\sqrt{b^{2} \left(2 a + b \left(\frac{\left(- a + b\right)^{2} \left(a + b\right)^{2}}{\left(a^{2} + b^{2}\right)^{2}} - 1\right) + \left(- \frac{- a^{2} + b^{2}}{a^{2} + b^{2}} + 1\right)^{2}\right)}$$

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       /    /             2                                  \ 
      /     |/     2    2\            /           2        2\| 
     /    2 ||    b  - a |            |    (a + b) *(b - a) || 
    /    b *||1 - -------|  + 2*a - b*|1 - -----------------|| 
   /        ||     2    2|            |                 2   || 
  /         |\    a  + b /            |        / 2    2\    || 
\/          \                         \        \a  + b /    //

sqrt(b^2*((1 - (b^2 - a^2)/(a^2 + b^2))^2 + 2*a - b*(1 - (a + b)^2*(b - a)^2/(a^2 + b^2)^2)))

Combinatorics [src]

            __________________________________________
           /    2 / 4    4      3        3      2  2\ 
  ___     /  a*b *\a  + b  + 2*a  - 2*a*b  + 2*a *b / 
\/ 2 *   /   ---------------------------------------- 
        /                4    4      2  2             
      \/                a  + b  + 2*a *b

$$\sqrt{2} \sqrt{\frac{a b^{2} \left(a^{4} + 2 a^{3} + 2 a^{2} b^{2} - 2 a b^{3} + b^{4}\right)}{a^{4} + 2 a^{2} b^{2} + b^{4}}}$$

sqrt(2)*sqrt(a*b^2*(a^4 + b^4 + 2*a^3 - 2*a*b^3 + 2*a^2*b^2)/(a^4 + b^4 + 2*a^2*b^2))

Trigonometric part [src]

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       /    /             2                                  \ 
      /     |/     2    2\            /           2        2\| 
     /    2 ||    b  - a |            |    (a + b) *(b - a) || 
    /    b *||1 - -------|  + 2*a - b*|1 - -----------------|| 
   /        ||     2    2|            |                 2   || 
  /         |\    a  + b /            |        / 2    2\    || 
\/          \                         \        \a  + b /    //

sqrt(b^2*((1 - (b^2 - a^2)/(a^2 + b^2))^2 + 2*a - b*(1 - (a + b)^2*(b - a)^2/(a^2 + b^2)^2)))

Rational denominator [src]

           ________________________________________________________________________________________________
          /    /                                  2                                                      \ 
         /     |         2 /        2         2  \               2                2                      | 
        /    2 |/ 2    2\  |       a         b   |      / 2    2\        / 2    2\             2        2| 
       /    b *|\b  + a / *|1 + ------- - -------|  - b*\b  + a /  + 2*a*\b  + a /  + b*(a + b) *(a - b) | 
      /        |           |     2    2    2    2|                                                       | 
     /         \           \    b  + a    b  + a /                                                       / 
    /       ---------------------------------------------------------------------------------------------- 
   /                                                           2                                           
  /                                                   / 2    2\                                            
\/                                                    \b  + a /

$$\sqrt{\frac{b^{2} \left(2 a \left(a^{2} + b^{2}\right)^{2} + b \left(a - b\right)^{2} \left(a + b\right)^{2} - b \left(a^{2} + b^{2}\right)^{2} + \left(a^{2} + b^{2}\right)^{2} \left(\frac{a^{2}}{a^{2} + b^{2}} - \frac{b^{2}}{a^{2} + b^{2}} + 1\right)^{2}\right)}{\left(a^{2} + b^{2}\right)^{2}}}$$

sqrt(b^2*((b^2 + a^2)^2*(1 + a^2/(b^2 + a^2) - b^2/(b^2 + a^2))^2 - b*(b^2 + a^2)^2 + 2*a*(b^2 + a^2)^2 + b*(a + b)^2*(a - b)^2)/(b^2 + a^2)^2)

Common denominator [src]

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     /                    6                   7               4                    4  2                4  3                 2  4                2  5           2  2 
    /   2    3           b                   b             2*b          2         a *b                a *b               2*a *b              2*a *b         2*a *b  
   /   b  - b  + ----------------- + ----------------- - ------- + 2*a*b  + ----------------- + ----------------- - ----------------- - ----------------- + ------- 
  /               4    4      2  2    4    4      2  2    2    2             4    4      2  2    4    4      2  2    4    4      2  2    4    4      2  2    2    2 
\/               a  + b  + 2*a *b    a  + b  + 2*a *b    a  + b             a  + b  + 2*a *b    a  + b  + 2*a *b    a  + b  + 2*a *b    a  + b  + 2*a *b    a  + b

sqrt(b^2 - b^3 + b^6/(a^4 + b^4 + 2*a^2*b^2) + b^7/(a^4 + b^4 + 2*a^2*b^2) - 2*b^4/(a^2 + b^2) + 2*a*b^2 + a^4*b^2/(a^4 + b^4 + 2*a^2*b^2) + a^4*b^3/(a^4 + b^4 + 2*a^2*b^2) - 2*a^2*b^4/(a^4 + b^4 + 2*a^2*b^2) - 2*a^2*b^5/(a^4 + b^4 + 2*a^2*b^2) + 2*a^2*b^2/(a^2 + b^2))

Powers [src]

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       /    /             2                                  \ 
      /     |/     2    2\            /           2        2\| 
     /    2 ||    a  - b |            |    (a + b) *(b - a) || 
    /    b *||1 + -------|  + 2*a - b*|1 - -----------------|| 
   /        ||     2    2|            |                 2   || 
  /         |\    a  + b /            |        / 2    2\    || 
\/          \                         \        \a  + b /    //

$$\sqrt{b^{2} \left(2 a - b \left(- \frac{\left(- a + b\right)^{2} \left(a + b\right)^{2}}{\left(a^{2} + b^{2}\right)^{2}} + 1\right) + \left(\frac{a^{2} - b^{2}}{a^{2} + b^{2}} + 1\right)^{2}\right)}$$

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       /    /             2                                  \ 
      /     |/     2    2\            /           2        2\| 
     /    2 ||    b  - a |            |    (a + b) *(b - a) || 
    /    b *||1 - -------|  + 2*a - b*|1 - -----------------|| 
   /        ||     2    2|            |                 2   || 
  /         |\    a  + b /            |        / 2    2\    || 
\/          \                         \        \a  + b /    //

sqrt(b^2*((1 - (b^2 - a^2)/(a^2 + b^2))^2 + 2*a - b*(1 - (a + b)^2*(b - a)^2/(a^2 + b^2)^2)))

Combining rational expressions [src]

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       /    /         /         2                    \                2\ 
      /   2 |   4     |/ 2    2\           2        2|       / 2    2\ | 
     /   b *\4*a  - b*\\a  + b /  - (a + b) *(b - a) / + 2*a*\a  + b / / 
    /    --------------------------------------------------------------- 
   /                                         2                           
  /                                 / 2    2\                            
\/                                  \a  + b /

$$\sqrt{\frac{b^{2} \left(4 a^{4} + 2 a \left(a^{2} + b^{2}\right)^{2} - b \left(- \left(- a + b\right)^{2} \left(a + b\right)^{2} + \left(a^{2} + b^{2}\right)^{2}\right)\right)}{\left(a^{2} + b^{2}\right)^{2}}}$$

sqrt(b^2*(4*a^4 - b*((a^2 + b^2)^2 - (a + b)^2*(b - a)^2) + 2*a*(a^2 + b^2)^2)/(a^2 + b^2)^2)

Other calculators

How to use it?

Identical expressions

Similar expressions

Least common denominator sqrt(b^2*((1-(b^2-a^2)/(b^2+a^2))^2+(2*a-b*sqrt(1-((b-a)*(b+a)/(b^2+a^2))^2)^2)))

The solution

Least common denominator sqrt(b^2((1-(b^2-a^2)/(b^2+a^2))^2+(2a-bsqrt(1-((b-a)(b+a)/(b^2+a^2))^2)^2)))