Problem of the Week

Updated at Aug 20, 2018 4:47 PM

How would you solve (v+23)26=23\frac{{(\frac{v+2}{3})}^{2}}{6}=\frac{2}{3}?

Below is the solution.



(v+23)26=23\frac{{(\frac{v+2}{3})}^{2}}{6}=\frac{2}{3}

1
Use Division Distributive Property: (xy)a=xaya{(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}.
(v+2)2326=23\frac{\frac{{(v+2)}^{2}}{{3}^{2}}}{6}=\frac{2}{3}

2
Simplify  32{3}^{2}  to  99.
(v+2)296=23\frac{\frac{{(v+2)}^{2}}{9}}{6}=\frac{2}{3}

3
Simplify  (v+2)296\frac{\frac{{(v+2)}^{2}}{9}}{6}  to  (v+2)29×6\frac{{(v+2)}^{2}}{9\times 6}.
(v+2)29×6=23\frac{{(v+2)}^{2}}{9\times 6}=\frac{2}{3}

4
Simplify  9×69\times 6  to  5454.
(v+2)254=23\frac{{(v+2)}^{2}}{54}=\frac{2}{3}

5
Multiply both sides by 5454.
(v+2)2=23×54{(v+2)}^{2}=\frac{2}{3}\times 54

6
Use this rule: ab×c=acb\frac{a}{b} \times c=\frac{ac}{b}.
(v+2)2=2×543{(v+2)}^{2}=\frac{2\times 54}{3}

7
Simplify  2×542\times 54  to  108108.
(v+2)2=1083{(v+2)}^{2}=\frac{108}{3}

8
Simplify  1083\frac{108}{3}  to  3636.
(v+2)2=36{(v+2)}^{2}=36

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

10
Since 6×6=366\times 6=36, the square root of 3636 is 66.
v+2=±6v+2=\pm 6

11
Break down the problem into these 2 equations.
v+2=6v+2=6
v+2=−6v+2=-6

12
Solve the 1st equation: v+2=6v+2=6.
v=4v=4

13
Solve the 2nd equation: v+2=−6v+2=-6.
v=−8v=-8

14
Collect all solutions.
v=4,−8v=4,-8

Done
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