Problem of the Week

Updated at Aug 20, 2018 4:47 PM

How would you solve \(\frac{{(\frac{v+2}{3})}^{2}}{6}=\frac{2}{3}\)?

Below is the solution.



\[\frac{{(\frac{v+2}{3})}^{2}}{6}=\frac{2}{3}\]

1
Use Division Distributive Property: \({(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}\).
\[\frac{\frac{{(v+2)}^{2}}{{3}^{2}}}{6}=\frac{2}{3}\]

2
Simplify  \({3}^{2}\)  to  \(9\).
\[\frac{\frac{{(v+2)}^{2}}{9}}{6}=\frac{2}{3}\]

3
Simplify  \(\frac{\frac{{(v+2)}^{2}}{9}}{6}\)  to  \(\frac{{(v+2)}^{2}}{9\times 6}\).
\[\frac{{(v+2)}^{2}}{9\times 6}=\frac{2}{3}\]

4
Simplify  \(9\times 6\)  to  \(54\).
\[\frac{{(v+2)}^{2}}{54}=\frac{2}{3}\]

5
Multiply both sides by \(54\).
\[{(v+2)}^{2}=\frac{2}{3}\times 54\]

6
Use this rule: \(\frac{a}{b} \times c=\frac{ac}{b}\).
\[{(v+2)}^{2}=\frac{2\times 54}{3}\]

7
Simplify  \(2\times 54\)  to  \(108\).
\[{(v+2)}^{2}=\frac{108}{3}\]

8
Simplify  \(\frac{108}{3}\)  to  \(36\).
\[{(v+2)}^{2}=36\]

9
Take the square root of both sides.
\[v+2=\pm \sqrt{36}\]

10
Since \(6\times 6=36\), the square root of \(36\) is \(6\).
\[v+2=\pm 6\]

11
Break down the problem into these 2 equations.
\[v+2=6\]
\[v+2=-6\]

12
Solve the 1st equation: \(v+2=6\).
\[v=4\]

13
Solve the 2nd equation: \(v+2=-6\).
\[v=-8\]

14
Collect all solutions.
\[v=4,-8\]

Done
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