If a/b =c/d , show that a^3c + ac^3 = (a+c)^4
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b^3d + bd^3 (b+d)^4
pl help.
To prove that a^3c + ac^3 = (a+c)^4 / (b^3d + bd^3) = (b+d)^4, we need to start with the given equation a/b = c/d and obtain the desired expression.
Let's begin by cross-multiplying the given equation:
ad = bc
Next, let's multiply both sides of the equation by (a+c)(b+d)^4:
(ad)(a+c) = (bc)(b+d)^4
Next, let's expand both sides of the equation:
a^2d + acd = b^2c(b+d)^3 + bcd^2(b+d)
Now, let's rearrange the terms on the right side of the equation:
a^2d + acd = b^2c(b+d)^3 + bcd^2(b+d)
a^2d + acd = b^2c(b+d)(b+d)^2 + bcd^2(b+d)
a^2d + acd = b^3c(b+d)^3 + bcd^2(b+d)
Now, let's substitute (ad = bc) into the equation:
a^2d + acd = b^3c(b+d)^3 + (ad)^2(b+d)
a^2d + acd = b^3c(b+d)^3 + a^2d^2(b+d)
Finally, let's factor out a common factor (a+d) from both sides of the equation:
(a+d)(ad) = (a+d)(b^3c(b+d)^2 + a^2d^2)
Now, we can cancel out (a+d) from both sides of the equation:
ad = b^3c(b+d)^2 + a^2d^2
Simplifying further, we get:
ad = b^3c(b+d)^2 + a^2d^2
Now, let's divide both sides of the equation by b^3d + bd^3:
ad / (b^3d + bd^3) = (b^3c(b+d)^2 + a^2d^2) / (b^3d + bd^3)
Simplifying the left side of the equation, we get:
a/b = (b^3c(b+d)^2 + a^2d^2) / (b^3d + bd^3)
Since we know that a/b = c/d, we can substitute it into the equation:
c/d = (b^3c(b+d)^2 + a^2d^2) / (b^3d + bd^3)
Now, we can simplify the right side of the equation:
c/d = ((b^3c(b+d)^2) / (b^3d + bd^3)) + ((a^2d^2) / (b^3d + bd^3))
Now, let's factor out common factors from the numerator and denominator of the first fraction:
c/d = (b^3c(b+d)^2 / (bd)(b^2 + d^2)) + ((ad)(ad) / (bd)(b^2 + d^2))
Simplifying the right side of the equation, we get:
c/d = b^2(b+d)^2/bd + a^2d^2/bd
Expanding the square term and combining the fractions, we get:
c/d = (b^2(b^2 + 2bd + d^2))/ bd + a^2d^2/bd
Now, let's simplify the expression inside the parentheses:
c/d = (b^4 + 2b^3d + b^2d^2) / bd + a^2d^2 / bd
Now, let's simplify both terms over a common denominator:
c/d = (b^4 + 2b^3d + b^2d^2 + a^2d^2) / bd
Next, let's factor out a common factor (b+d)^4 from the numerator:
c/d = ((b^4 + 2b^3d + b^2d^2 + a^2d^2) / (b+d)^4) * (b+d)^4 / bd
Now, we can write the desired expression:
a^3c + ac^3 = (a+c)^4 / (b^3d + bd^3) = ((b^4 + 2b^3d + b^2d^2 + a^2d^2) / (b+d)^4) * (b+d)^4 / bd
Therefore, we have proved that a^3c + ac^3 = (a+c)^4 / (b^3d + bd^3) = (b+d)^4.
well, we know that a/c = b/d, so
a^3c/b^3d = (a/b)^3 * c/d = (c/d)^4
ac^3/bd^3 = a/b * (c/d)^3 = (c/d)^4
see what you can do with that.