Problem of the Week

Updated at Aug 24, 2020 11:12 AM

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

Below is the solution.



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

1
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\frac{4x}{5\times \frac{{x}^{2}}{{5}^{2}}}=5\]

2
Simplify  \({5}^{2}\)  to  \(25\).
\[\frac{4x}{5\times \frac{{x}^{2}}{25}}=5\]

3
Simplify  \(5\times \frac{{x}^{2}}{25}\)  to  \(\frac{{x}^{2}}{5}\).
\[\frac{4x}{\frac{{x}^{2}}{5}}=5\]

4
Invert and multiply.
\[4x\times \frac{5}{{x}^{2}}=5\]

5
Simplify  \(4x\times \frac{5}{{x}^{2}}\)  to  \(\frac{20x}{{x}^{2}}\).
\[\frac{20x}{{x}^{2}}=5\]

6
Use Quotient Rule: \(\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}\).
\[20{x}^{1-2}=5\]

7
Simplify  \(1-2\)  to  \(-1\).
\[20{x}^{-1}=5\]

8
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[20\times \frac{1}{x}=5\]

9
Simplify  \(20\times \frac{1}{x}\)  to  \(\frac{20}{x}\).
\[\frac{20}{x}=5\]

10
Multiply both sides by \(x\).
\[20=5x\]

11
Divide both sides by \(5\).
\[\frac{20}{5}=x\]

12
Simplify  \(\frac{20}{5}\)  to  \(4\).
\[4=x\]

13
Switch sides.
\[x=4\]

Done
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