Solve for a (abab)/(cdcd)=(ab)/(cd)

Math
ababcdcd=abcd
Multiply both sides of the equation by cdcd.
abab=abcd⋅(cdcd)
Simplify abab.
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Multiply a by a by adding the exponents.
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Move a.
a⋅ab⋅b=abcd⋅(cdcd)
Multiply a by a.
a2b⋅b=abcd⋅(cdcd)
a2b⋅b=abcd⋅(cdcd)
Multiply b by b by adding the exponents.
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Move b.
a2(b⋅b)=abcd⋅(cdcd)
Multiply b by b.
a2b2=abcd⋅(cdcd)
a2b2=abcd⋅(cdcd)
a2b2=abcd⋅(cdcd)
Cancel the common factor of cd.
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Factor cd out of cdcd.
a2b2=abcd⋅(cd(cd))
Cancel the common factor.
a2b2=abcd⋅(cd(cd))
Rewrite the expression.
a2b2=ab⋅(cd)
a2b2=abcd
Subtract abcd from both sides of the equation.
a2b2-abcd=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=b2, b=-bcd, and c=0 into the quadratic formula and solve for a.
bcd±(-bcd)2-4⋅(b2⋅0)2b2
Simplify.
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Simplify the numerator.
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Rewrite 4⋅(b2⋅0) as (2⋅(b⋅0))2.
a=bcd±(-bcd)2-(2⋅(b⋅0))22b2
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=-bcd and b=2⋅(b⋅0).
a=bcd±(-bcd+2⋅(b⋅0))(-bcd-(2⋅(b⋅0)))2b2
Simplify.
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Factor b out of -bcd+2⋅(b⋅0).
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Factor b out of -bcd.
a=bcd±(b(-1cd)+2⋅(b⋅0))(-bcd-(2⋅(b⋅0)))2b2
Factor b out of 2⋅(b⋅0).
a=bcd±(b(-1cd)+b(2⋅(0)))(-bcd-(2⋅(b⋅0)))2b2
Factor b out of b(-1cd)+b(2⋅(0)).
a=bcd±b(-1cd+2⋅(0))(-bcd-(2⋅(b⋅0)))2b2
a=bcd±b(-1cd+2⋅(0))(-bcd-(2⋅(b⋅0)))2b2
Rewrite -1c as -c.
a=bcd±b(-cd+2⋅(0))(-bcd-(2⋅(b⋅0)))2b2
Multiply 2 by 0.
a=bcd±b(-cd+0)(-bcd-(2⋅(b⋅0)))2b2
Add -cd and 0.
a=bcd±b⋅(-1cd(-bcd-(2⋅(b⋅0))))2b2
Factor out negative.
a=bcd±-bcd(-bcd-(2⋅(b⋅0)))2b2
Combine exponents.
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Multiply b by 0.
a=bcd±-bcd(-bcd-(2⋅0))2b2
Multiply 2 by 0.
a=bcd±-bcd(-bcd-0)2b2
Multiply -1 by 0.
a=bcd±-bcd(-bcd+0)2b2
a=bcd±-bcd(-bcd+0)2b2
a=bcd±-bcd(-bcd+0)2b2
Add -bcd and 0.
a=bcd±-bcd(-bcd)2b2
Multiply -1 by -1.
a=bcd±1bcd(bc)d2b2
Multiply b by 1.
a=bcd±bcd(bc)d2b2
Raise b to the power of 1.
a=bcd±b⋅bcdcd2b2
Raise b to the power of 1.
a=bcd±b⋅bcdcd2b2
Use the power rule aman=am+n to combine exponents.
a=bcd±b1+1cdcd2b2
Add 1 and 1.
a=bcd±b2cdcd2b2
Raise c to the power of 1.
a=bcd±b2(c⋅c)d⋅d2b2
Raise c to the power of 1.
a=bcd±b2(c⋅c)d⋅d2b2
Use the power rule aman=am+n to combine exponents.
a=bcd±b2c1+1d⋅d2b2
Add 1 and 1.
a=bcd±b2c2d⋅d2b2
Raise d to the power of 1.
a=bcd±b2c2(d⋅d)2b2
Raise d to the power of 1.
a=bcd±b2c2(d⋅d)2b2
Use the power rule aman=am+n to combine exponents.
a=bcd±b2c2d1+12b2
Add 1 and 1.
a=bcd±b2c2d22b2
Rewrite b2c2d2 as (bcd)2.
a=bcd±(bcd)22b2
Pull terms out from under the radical, assuming positive real numbers.
a=bcd±bcd2b2
a=bcd±bcd2b2
Simplify bcd±bcd2b2.
a=cd±cd2b
a=cd±cd2b
The final answer is the combination of both solutions.
a=cdb
a=0
Solve for a (abab)/(cdcd)=(ab)/(cd)

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