Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
sqrt((l1-v1*((v1*l1+v2*l2)/(v2^2+v1^2)))^2+(l2-v2*((v1*l1+v2*l2)/(v2^2+v1^2)))^2)
How do you sqrt((l1-v1*((v1*l1+v2*l2)/(v2^2+v1^2)))^2+(l2-v2*((v1*l1+v2*l2)/(v2^2+v1^2)))^2) in partial fractions?
The solution
You have entered
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/ 2 2
/ / v1*l1 + v2*l2\ / v1*l1 + v2*l2\
/ |l1 - v1*-------------| + |l2 - v2*-------------|
/ | 2 2 | | 2 2 |
\/ \ v2 + v1 / \ v2 + v1 /
$$\sqrt{\left(l_{1} - v_{1} \frac{l_{1} v_{1} + l_{2} v_{2}}{v_{1}^{2} + v_{2}^{2}}\right)^{2} + \left(l_{2} - v_{2} \frac{l_{1} v_{1} + l_{2} v_{2}}{v_{1}^{2} + v_{2}^{2}}\right)^{2}}$$
sqrt((l1 - v1*(v1*l1 + v2*l2)/(v2^2 + v1^2))^2 + (l2 - v2*(v1*l1 + v2*l2)/(v2^2 + v1^2))^2)
Fraction decomposition
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sqrt(l1^2 + l2^2 + l1^2*v1^4/(v1^4 + v2^4 + 2*v1^2*v2^2) + l2^2*v2^4/(v1^4 + v2^4 + 2*v1^2*v2^2) - 2*l1^2*v1^2/(v2^2 + v1^2) - 2*l2^2*v2^2/(v2^2 + v1^2) + l1^2*v1^2*v2^2/(v1^4 + v2^4 + 2*v1^2*v2^2) + l2^2*v1^2*v2^2/(v1^4 + v2^4 + 2*v1^2*v2^2) - 4*l1*l2*v1*v2/(v2^2 + v1^2) + 2*l1*l2*v1*v2^3/(v1^4 + v2^4 + 2*v1^2*v2^2) + 2*l1*l2*v2*v1^3/(v1^4 + v2^4 + 2*v1^2*v2^2))
$$\sqrt{\frac{l_{1}^{2} v_{1}^{4}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} + \frac{l_{1}^{2} v_{1}^{2} v_{2}^{2}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{2 l_{1}^{2} v_{1}^{2}}{v_{1}^{2} + v_{2}^{2}} + l_{1}^{2} + \frac{2 l_{1} l_{2} v_{1}^{3} v_{2}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} + \frac{2 l_{1} l_{2} v_{1} v_{2}^{3}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{4 l_{1} l_{2} v_{1} v_{2}}{v_{1}^{2} + v_{2}^{2}} + \frac{l_{2}^{2} v_{1}^{2} v_{2}^{2}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} + \frac{l_{2}^{2} v_{2}^{4}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{2 l_{2}^{2} v_{2}^{2}}{v_{1}^{2} + v_{2}^{2}} + l_{2}^{2}}$$
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/ 2 4 2 4 2 2 2 2 2 2 2 2 2 2 3 3
/ 2 2 l1 *v1 l2 *v2 2*l1 *v1 2*l2 *v2 l1 *v1 *v2 l2 *v1 *v2 4*l1*l2*v1*v2 2*l1*l2*v1*v2 2*l1*l2*v2*v1
/ l1 + l2 + --------------------- + --------------------- - --------- - --------- + --------------------- + --------------------- - ------------- + --------------------- + ---------------------
/ 4 4 2 2 4 4 2 2 2 2 2 2 4 4 2 2 4 4 2 2 2 2 4 4 2 2 4 4 2 2
\/ v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2 v2 + v1 v2 + v1 v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2 v2 + v1 v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2
General simplification
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/ 2 2
/ / 2 \ / 2 \
/ \l1*v2 - l2*v1*v2/ \l2*v1 - l1*v1*v2/
/ -------------------- + --------------------
/ 2 2
/ / 2 2\ / 2 2\
\/ \v1 + v2 / \v1 + v2 /
$$\sqrt{\frac{\left(l_{1} v_{2}^{2} - l_{2} v_{1} v_{2}\right)^{2}}{\left(v_{1}^{2} + v_{2}^{2}\right)^{2}} + \frac{\left(- l_{1} v_{1} v_{2} + l_{2} v_{1}^{2}\right)^{2}}{\left(v_{1}^{2} + v_{2}^{2}\right)^{2}}}$$
sqrt((l1*v2^2 - l2*v1*v2)^2/(v1^2 + v2^2)^2 + (l2*v1^2 - l1*v1*v2)^2/(v1^2 + v2^2)^2)
Rational denominator
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/ 2 2
/ / 2 \ / 2 \
/ | l1*v1 l2*v1*v2| | l2*v2 l1*v1*v2|
/ |-l1 + --------- + ---------| + |-l2 + --------- + ---------|
/ | 2 2 2 2| | 2 2 2 2|
\/ \ v2 + v1 v2 + v1 / \ v2 + v1 v2 + v1 /
$$\sqrt{\left(\frac{l_{1} v_{1}^{2}}{v_{1}^{2} + v_{2}^{2}} - l_{1} + \frac{l_{2} v_{1} v_{2}}{v_{1}^{2} + v_{2}^{2}}\right)^{2} + \left(\frac{l_{1} v_{1} v_{2}}{v_{1}^{2} + v_{2}^{2}} + \frac{l_{2} v_{2}^{2}}{v_{1}^{2} + v_{2}^{2}} - l_{2}\right)^{2}}$$
sqrt((-l1 + l1*v1^2/(v2^2 + v1^2) + l2*v1*v2/(v2^2 + v1^2))^2 + (-l2 + l2*v2^2/(v2^2 + v1^2) + l1*v1*v2/(v2^2 + v1^2))^2)
Common denominator
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/ 2 4 2 4 2 2 2 2 2 2 2 2 2 2 3 3
/ 2 2 l1 *v1 l2 *v2 2*l1 *v1 2*l2 *v2 l1 *v1 *v2 l2 *v1 *v2 4*l1*l2*v1*v2 2*l1*l2*v1*v2 2*l1*l2*v2*v1
/ l1 + l2 + --------------------- + --------------------- - --------- - --------- + --------------------- + --------------------- - ------------- + --------------------- + ---------------------
/ 4 4 2 2 4 4 2 2 2 2 2 2 4 4 2 2 4 4 2 2 2 2 4 4 2 2 4 4 2 2
\/ v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2 v1 + v2 v1 + v2 v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2 v1 + v2 v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2
$$\sqrt{\frac{l_{1}^{2} v_{1}^{4}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} + \frac{l_{1}^{2} v_{1}^{2} v_{2}^{2}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{2 l_{1}^{2} v_{1}^{2}}{v_{1}^{2} + v_{2}^{2}} + l_{1}^{2} + \frac{2 l_{1} l_{2} v_{1}^{3} v_{2}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} + \frac{2 l_{1} l_{2} v_{1} v_{2}^{3}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{4 l_{1} l_{2} v_{1} v_{2}}{v_{1}^{2} + v_{2}^{2}} + \frac{l_{2}^{2} v_{1}^{2} v_{2}^{2}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} + \frac{l_{2}^{2} v_{2}^{4}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{2 l_{2}^{2} v_{2}^{2}}{v_{1}^{2} + v_{2}^{2}} + l_{2}^{2}}$$
sqrt(l1^2 + l2^2 + l1^2*v1^4/(v1^4 + v2^4 + 2*v1^2*v2^2) + l2^2*v2^4/(v1^4 + v2^4 + 2*v1^2*v2^2) - 2*l1^2*v1^2/(v1^2 + v2^2) - 2*l2^2*v2^2/(v1^2 + v2^2) + l1^2*v1^2*v2^2/(v1^4 + v2^4 + 2*v1^2*v2^2) + l2^2*v1^2*v2^2/(v1^4 + v2^4 + 2*v1^2*v2^2) - 4*l1*l2*v1*v2/(v1^2 + v2^2) + 2*l1*l2*v1*v2^3/(v1^4 + v2^4 + 2*v1^2*v2^2) + 2*l1*l2*v2*v1^3/(v1^4 + v2^4 + 2*v1^2*v2^2))
Assemble expression
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/ 2 2
/ / v1*(l1*v1 + l2*v2)\ / v2*(l1*v1 + l2*v2)\
/ |l1 - ------------------| + |l2 - ------------------|
/ | 2 2 | | 2 2 |
\/ \ v1 + v2 / \ v1 + v2 /
$$\sqrt{\left(l_{1} - \frac{v_{1} \left(l_{1} v_{1} + l_{2} v_{2}\right)}{v_{1}^{2} + v_{2}^{2}}\right)^{2} + \left(l_{2} - \frac{v_{2} \left(l_{1} v_{1} + l_{2} v_{2}\right)}{v_{1}^{2} + v_{2}^{2}}\right)^{2}}$$
sqrt((l1 - v1*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2 + (l2 - v2*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2)
Trigonometric part
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/ 2 2
/ / v1*(l1*v1 + l2*v2)\ / v2*(l1*v1 + l2*v2)\
/ |l1 - ------------------| + |l2 - ------------------|
/ | 2 2 | | 2 2 |
\/ \ v1 + v2 / \ v1 + v2 /
$$\sqrt{\left(l_{1} - \frac{v_{1} \left(l_{1} v_{1} + l_{2} v_{2}\right)}{v_{1}^{2} + v_{2}^{2}}\right)^{2} + \left(l_{2} - \frac{v_{2} \left(l_{1} v_{1} + l_{2} v_{2}\right)}{v_{1}^{2} + v_{2}^{2}}\right)^{2}}$$
sqrt((l1 - v1*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2 + (l2 - v2*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2)
Combinatorics
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/ 2 4 2 4 2 2 2 2 2 2 3 3
/ 2 2 l1 *v1 l2 *v2 l1 *v1 *v2 l2 *v1 *v2 2*l1*l2*v1*v2 2*l1*l2*v2*v1
/ l1 + l2 - --------------------- - --------------------- - --------------------- - --------------------- - --------------------- - ---------------------
/ 4 4 2 2 4 4 2 2 4 4 2 2 4 4 2 2 4 4 2 2 4 4 2 2
\/ v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2 v1 + v2 + 2*v1 *v2
$$\sqrt{- \frac{l_{1}^{2} v_{1}^{4}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{l_{1}^{2} v_{1}^{2} v_{2}^{2}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} + l_{1}^{2} - \frac{2 l_{1} l_{2} v_{1}^{3} v_{2}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{2 l_{1} l_{2} v_{1} v_{2}^{3}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{l_{2}^{2} v_{1}^{2} v_{2}^{2}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} - \frac{l_{2}^{2} v_{2}^{4}}{v_{1}^{4} + 2 v_{1}^{2} v_{2}^{2} + v_{2}^{4}} + l_{2}^{2}}$$
sqrt(l1^2 + l2^2 - l1^2*v1^4/(v1^4 + v2^4 + 2*v1^2*v2^2) - l2^2*v2^4/(v1^4 + v2^4 + 2*v1^2*v2^2) - l1^2*v1^2*v2^2/(v1^4 + v2^4 + 2*v1^2*v2^2) - l2^2*v1^2*v2^2/(v1^4 + v2^4 + 2*v1^2*v2^2) - 2*l1*l2*v1*v2^3/(v1^4 + v2^4 + 2*v1^2*v2^2) - 2*l1*l2*v2*v1^3/(v1^4 + v2^4 + 2*v1^2*v2^2))
Numerical answer
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((l1 - v1*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2 + (l2 - v2*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2)^0.5
((l1 - v1*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2 + (l2 - v2*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2)^0.5
Combining rational expressions
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/ 2 2
/ / / 2 2\ \ / / 2 2\ \
/ \l1*\v1 + v2 / - v1*(l1*v1 + l2*v2)/ + \l2*\v1 + v2 / - v2*(l1*v1 + l2*v2)/
/ -------------------------------------------------------------------------------
/ 2
/ / 2 2\
\/ \v1 + v2 /
$$\sqrt{\frac{\left(l_{1} \left(v_{1}^{2} + v_{2}^{2}\right) - v_{1} \left(l_{1} v_{1} + l_{2} v_{2}\right)\right)^{2} + \left(l_{2} \left(v_{1}^{2} + v_{2}^{2}\right) - v_{2} \left(l_{1} v_{1} + l_{2} v_{2}\right)\right)^{2}}{\left(v_{1}^{2} + v_{2}^{2}\right)^{2}}}$$
sqrt(((l1*(v1^2 + v2^2) - v1*(l1*v1 + l2*v2))^2 + (l2*(v1^2 + v2^2) - v2*(l1*v1 + l2*v2))^2)/(v1^2 + v2^2)^2)
Powers
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/ 2 2
/ / v1*(l1*v1 + l2*v2)\ / v2*(l1*v1 + l2*v2)\
/ |l1 - ------------------| + |l2 - ------------------|
/ | 2 2 | | 2 2 |
\/ \ v1 + v2 / \ v1 + v2 /
$$\sqrt{\left(l_{1} - \frac{v_{1} \left(l_{1} v_{1} + l_{2} v_{2}\right)}{v_{1}^{2} + v_{2}^{2}}\right)^{2} + \left(l_{2} - \frac{v_{2} \left(l_{1} v_{1} + l_{2} v_{2}\right)}{v_{1}^{2} + v_{2}^{2}}\right)^{2}}$$
sqrt((l1 - v1*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2 + (l2 - v2*(l1*v1 + l2*v2)/(v1^2 + v2^2))^2)