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

Simplify: (xaxb)b+c−a(xcxa)c+a−b(xaxb)a+b−c \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}

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Find derivatives of using multiple formulaeright chevron icon

Full solution

Q. Simplify: (xaxb)b+c−a(xcxa)c+a−b(xaxb)a+b−c \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}
  1. Simplify fractions using exponents: Step 11: Simplify each fraction using the properties of exponents.\newline(xaxb)=xa−b\left(\frac{x^{a}}{x^{b}}\right) = x^{a-b}\newline(xcxa)=xc−a\left(\frac{x^{c}}{x^{a}}\right) = x^{c-a}\newline(xaxb)=xa−b\left(\frac{x^{a}}{x^{b}}\right) = x^{a-b}
  2. Apply exponents to simplified fractions: Step 22: Apply the exponents to the simplified fractions.\newline(x(a−b))(b+c−a)=x((a−b)∗(b+c−a))(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))(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))(x^{(a-b)})^{(a+b-c)} = x^{((a-b)*(a+b-c))}
  3. Multiply results from Step 22: Step 33: Multiply the results from Step 22. x((a−b)∗(b+c−a)+(c−a)∗(c+a−b)+(a−b)∗(a+b−c))x^{((a-b)*(b+c-a) + (c-a)*(c+a-b) + (a-b)*(a+b-c))}
  4. Expand and simplify the exponent: Step 44: Expand and simplify the exponent.\newline(a−b)⋅(b+c−a)=ab−a2+bc−ba,(a-b)\cdot(b+c-a) = ab - a^2 + bc - ba,\newline(c−a)⋅(c+a−b)=c2−ca+ac−ab,(c-a)\cdot(c+a-b) = c^2 - ca + ac - ab,\newline(a−b)⋅(a+b−c)=a2−ab+ab−bc.(a-b)\cdot(a+b-c) = a^2 - ab + ab - bc.\newlineSumming these gives: ab−a2+bc−ba+c2−ca+ac−ab+a2−ab+ab−bc=a2+c2−a2−ca+ac−ca.ab - a^2 + bc - ba + c^2 - ca + ac - ab + a^2 - ab + ab - bc = a^2 + c^2 - a^2 - ca + ac - ca.\newlineSimplifies to,\newlinebk+c2−ca+ac−b2bk + c^2 - ca + ac - b^2.

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