Problem of the Week

Updated at Aug 24, 2020 11:12 AM

How can we solve the equation 4x5(x5)2=5\frac{4x}{5{(\frac{x}{5})}^{2}}=5?

Below is the solution.



4x5(x5)2=5\frac{4x}{5{(\frac{x}{5})}^{2}}=5

1
Use Division Distributive Property: (xy)a=xaya{(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}.
4x5×x252=5\frac{4x}{5\times \frac{{x}^{2}}{{5}^{2}}}=5

2
Simplify  52{5}^{2}  to  2525.
4x5×x225=5\frac{4x}{5\times \frac{{x}^{2}}{25}}=5

3
Simplify  5×x2255\times \frac{{x}^{2}}{25}  to  x25\frac{{x}^{2}}{5}.
4xx25=5\frac{4x}{\frac{{x}^{2}}{5}}=5

4
Invert and multiply.
4x×5x2=54x\times \frac{5}{{x}^{2}}=5

5
Simplify  4x×5x24x\times \frac{5}{{x}^{2}}  to  20xx2\frac{20x}{{x}^{2}}.
20xx2=5\frac{20x}{{x}^{2}}=5

6
Use Quotient Rule: xaxb=xa−b\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}.
20x1−2=520{x}^{1-2}=5

7
Simplify  1−21-2  to  −1-1.
20x−1=520{x}^{-1}=5

8
Use Negative Power Rule: x−a=1xa{x}^{-a}=\frac{1}{{x}^{a}}.
20×1x=520\times \frac{1}{x}=5

9
Simplify  20×1x20\times \frac{1}{x}  to  20x\frac{20}{x}.
20x=5\frac{20}{x}=5

10
Multiply both sides by xx.
20=5x20=5x

11
Divide both sides by 55.
205=x\frac{20}{5}=x

12
Simplify  205\frac{20}{5}  to  44.
4=x4=x

13
Switch sides.
x=4x=4

Done
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