V=h[( (a+b)/2)^2 + 1/3( (a-b)/2)^2]
showing all steps that it reduces to following:
v=h/3(a^2 +ab+b^2)
To simplify the given expression and reduce it to v = h/3(a^2 + ab + b^2), we will follow these steps:
Step 1: Expand the terms within the square brackets.
V = h[((a+b)/2)^2 + 1/3((a-b)/2)^2]
The squared term within the first set of brackets can be simplified as follows:
[(a+b)/2]^2 = (a+b)^2/4 = (a^2 + 2ab + b^2)/4
The squared term within the second set of brackets can be simplified similarly:
[(a-b)/2]^2 = (a-b)^2/4 = (a^2 - 2ab + b^2)/4
Step 2: Combine the expanded terms.
V = h[(a^2 + 2ab + b^2)/4 + (a^2 - 2ab + b^2)/4]
Step 3: Combine like terms.
Within the square brackets, we can add the terms that are being multiplied by h:
= h[(2a^2 + 2b^2)/4]
Step 4: Simplify the fraction within the brackets.
2a^2 + 2b^2 can be factored out the common factor of 2. We get:
= h[(2(a^2 + b^2))/4]
Step 5: Simplify the fraction h/4.
[(2(a^2 + b^2))/4] can be further simplified by dividing the numerator and denominator by 2:
= h[(a^2 + b^2)/2]
Step 6: Rewrite the expression as h/3 multiplied by (a^2 + ab + b^2).
If we multiply the fraction h/3 by (a^2 + ab + b^2)/(a^2 + b^2) (which equals 2/2), we can rewrite the expression as:
= h/3 * 2/2 * (a^2 + b^2)/2
Simplifying this, we get:
= h/3 * (2(a^2 + b^2))/2
= h/3 * (a^2 + b^2)
So, by following the steps above, we have successfully reduced the expression to v = h/3(a^2 + ab + b^2).