Example

3(r+2s)=2t-4
a⋅(b-2)=3b
3(r+2s)=2t-4
a⋅(b-2)=3b

Question

$$(x^{a-b})^{a-b}\times(x^{b-c})^{b-c}\times(x^{c-a})^{c-a}=1$$

Answer

$$x=1^(1/((a-b)^2+(b-c)^2+(c-a)^2))$$

Solution


Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[{x}^{(a-b)(a-b)}{({x}^{b-c})}^{b-c}{({x}^{c-a})}^{c-a}=1\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[{x}^{{(a-b)}^{2}}{({x}^{b-c})}^{b-c}{({x}^{c-a})}^{c-a}=1\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[{x}^{{(a-b)}^{2}}{x}^{(b-c)(b-c)}{({x}^{c-a})}^{c-a}=1\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[{x}^{{(a-b)}^{2}}{x}^{{(b-c)}^{2}}{({x}^{c-a})}^{c-a}=1\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[{x}^{{(a-b)}^{2}}{x}^{{(b-c)}^{2}}{x}^{(c-a)(c-a)}=1\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[{x}^{{(a-b)}^{2}}{x}^{{(b-c)}^{2}}{x}^{{(c-a)}^{2}}=1\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[{x}^{{(a-b)}^{2}+{(b-c)}^{2}+{(c-a)}^{2}}=1\]
Take the \(({(a-b)}^{2}+{(b-c)}^{2}+{(c-a)}^{2})\)th root of both sides.
\[x={\sqrt[(a-b)}^{2}+{(b-c)}^{2}+{(c-a)}^{2]{1}}\]
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