Que: Simplif ((x^(a))/(x^(b)))^(b+c-a)((x^(c))/(x^(a)))^(c+a-b)((x^(a))/(x^(b)))^(a+b-c)

Simplify fractions using exponents: Step 1: Simplify each fraction using the properties of exponents. ((x^(a))/(x^(b))) = x^(a-b) ((x^(c))/(x^(a))) = x^(c-a) ((x^(a))/(x^(b))) = x^(a-b)Apply exponents to simplified fractions: Step 2: Apply the exponents to the simplified fractions. (x^(a-b))^(b+c-a) = x^((a-b)*(b+c-a)) (x^(c-a))^(c+a-b) = x^((c-a)*(c+a-b)) (x^(a-b))^(a+b-c) = x^((a-b)*(a+b-c))Multiply results from Step 2: Step 3: Multiply the results from Step 2. x^((a-b)*(b+c-a) + (c-a)*(c+a-b) + (a-b)*(a+b-c))Expand and simplify the exponent: Step 4: Expand and simplify the exponent. (a-b)*(b+c-a) + (c-a)*(c+a-b) + (a-b)*(a+b-c) = a*b + a*c - a^2 - b^2 - b*c + b*a + c^2 + c*a - c*b - a*b - a*c + a^2 - b^2 - b*c + b*a + c^2 + c*a - c*b = -2b^2 + 2c^2

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Calculus

Find derivatives of using multiple formulae

Full solution

Q. Simplify:

\left(\frac{x^{a}}{x^{b}}\right)^{b+c-a}\left(\frac{x^{c}}{x^{a}}\right)^{c+a-b}\left(\frac{x^{a}}{x^{b}}\right)^{a+b-c}

Simplify fractions using exponents: Step $1$ : Simplify each fraction using the properties of exponents. $\newline$ $\left(\frac{x^{a}}{x^{b}}\right) = x^{a-b}$ $\newline$ $\left(\frac{x^{c}}{x^{a}}\right) = x^{c-a}$ $\newline$ $\left(\frac{x^{a}}{x^{b}}\right) = x^{a-b}$
Apply exponents to simplified fractions: Step $2$ : Apply the exponents to the simplified fractions. $\newline$ $(x^{(a-b)})^{(b+c-a)} = x^{((a-b)*(b+c-a))}$ $\newline$ $(x^{(c-a)})^{(c+a-b)} = x^{((c-a)*(c+a-b))}$ $\newline$ $(x^{(a-b)})^{(a+b-c)} = x^{((a-b)*(a+b-c))}$
Multiply results from Step $2$ : Step $3$ : Multiply the results from Step $2$ . $x^{((a-b)*(b+c-a) + (c-a)*(c+a-b) + (a-b)*(a+b-c))}$
Expand and simplify the exponent: Step $4$ : Expand and simplify the exponent. $\newline$ $(a-b)\cdot(b+c-a) = ab - a^2 + bc - ba,$ $\newline$ $(c-a)\cdot(c+a-b) = c^2 - ca + ac - ab,$ $\newline$ $(a-b)\cdot(a+b-c) = a^2 - ab + ab - bc.$ $\newline$ Summing these gives: $ab - a^2 + bc - ba + c^2 - ca + ac - ab + a^2 - ab + ab - bc = a^2 + c^2 - a^2 - ca + ac - ca.$ $\newline$ Simplifies to, $\newline$ $bk + c^2 - ca + ac - b^2$ .