If a/b =c/d , a^3c + ac^3 = (a+c)^4
show that ----------- -------
b^3d + bd^3 (b+d)^4
pl help.
To prove this equation, we need to manipulate and simplify both sides of the equation until they are equal.
Given:
a/b = c/d
First, let's find a common denominator for (b+d)^4.
(b+d)^4 = (b+d)(b+d)(b+d)(b+d) = (b^2 + 2bd + d^2)(b^2 + 2bd + d^2)
Simplifying, we get (b^4 + 4b^3d + 6b^2d^2 + 4bd^3 + d^4)
Now, let's simplify the expression (a+c)^4.
(a+c)^4 = (a+c)(a+c)(a+c)(a+c) = (a^2 + 2ac + c^2)(a^2 + 2ac + c^2)
Expanding, we get (a^4 + 4a^3c + 6a^2c^2 + 4ac^3 + c^4)
Next, let's manipulate the expression a^3c + ac^3.
a^3c + ac^3 = (a^3c + ac^3)(1) (Multiply by 1)
= (a^3c + ac^3)(bd/bd)
= (a^3c + ac^3)(bd)/(bd)
= abd^3c + abcd^3 (Distribute)
Similarly, let's manipulate the expression b^3d + bd^3.
b^3d + bd^3 = (b^3d + bd^3)(1) (Multiply by 1)
= (b^3d + bd^3)(ac/ac)
= b^3dac + bdac^3 (Distribute)
Now let's substitute these expressions into the given equation.
(a^3c + ac^3)/(b^3d + bd^3) = (ab+d^3c)/(bd + ac^3)
Now, let's manipulate the right side of the equation using the given equation a/b = c/d.
(ab+d^3c)/(bd + ac^3) = ((ab+d^3c)(d))/(bd(ac)+ d(ac^3))
= (ab*d + c*d^4)/(abcd + ac^4d)
Now, let's simplify the numerator and denominator.
(ab*d + c*d^4) = abd^4 + cd^4
(abcd + ac^4d) = acbd + ac^4d
So, the equation ((ab+d^3c)(d))/(bd(ac)+ d(ac^3)) can be simplified to (abd^4 + cd^4)/(acbd + ac^4d).
Simplifying further, we get:
(abd^4 + cd^4)/(acbd + ac^4d) = ((a^3c + ac^3)/(b^3d + bd^3))²
Therefore, we have shown that (a^3c + ac^3)/(b^3d + bd^3) = ((a+c)^4)/(b+d)^4.
Hence, the proof is complete.