Problem of the Week

Updated at Feb 15, 2021 1:23 PM

This week we have another equation problem:

How would you solve \(\frac{{(\frac{4x}{5})}^{2}}{5}=\frac{16}{5}\)?

Let's start!



\[\frac{{(\frac{4x}{5})}^{2}}{5}=\frac{16}{5}\]

1
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\frac{\frac{{(4x)}^{2}}{{5}^{2}}}{5}=\frac{16}{5}\]

2
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\frac{\frac{{4}^{2}{x}^{2}}{{5}^{2}}}{5}=\frac{16}{5}\]

3
Simplify  \({4}^{2}\)  to  \(16\).
\[\frac{\frac{16{x}^{2}}{{5}^{2}}}{5}=\frac{16}{5}\]

4
Simplify  \({5}^{2}\)  to  \(25\).
\[\frac{\frac{16{x}^{2}}{25}}{5}=\frac{16}{5}\]

5
Simplify  \(\frac{\frac{16{x}^{2}}{25}}{5}\)  to  \(\frac{16{x}^{2}}{25\times 5}\).
\[\frac{16{x}^{2}}{25\times 5}=\frac{16}{5}\]

6
Simplify  \(25\times 5\)  to  \(125\).
\[\frac{16{x}^{2}}{125}=\frac{16}{5}\]

7
Multiply both sides by \(125\).
\[16{x}^{2}=\frac{16}{5}\times 125\]

8
Use this rule: \(\frac{a}{b} \times c=\frac{ac}{b}\).
\[16{x}^{2}=\frac{16\times 125}{5}\]

9
Simplify  \(16\times 125\)  to  \(2000\).
\[16{x}^{2}=\frac{2000}{5}\]

10
Simplify  \(\frac{2000}{5}\)  to  \(400\).
\[16{x}^{2}=400\]

11
Divide both sides by \(16\).
\[{x}^{2}=\frac{400}{16}\]

12
Simplify  \(\frac{400}{16}\)  to  \(25\).
\[{x}^{2}=25\]

13
Take the square root of both sides.
\[x=\pm \sqrt{25}\]

14
Since \(5\times 5=25\), the square root of \(25\) is \(5\).
\[x=\pm 5\]

Done
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