Problem of the Week

Updated at Sep 23, 2024 9:22 AM

How would you solve the equation (52(4(2+n))2)=116(\frac{{5}^{2}}{{(4(2+n))}^{2}})=\frac{1}{16}?

Below is the solution.



(52(4(2+n))2)=116(\frac{{5}^{2}}{{(4(2+n))}^{2}})=\frac{1}{16}

1
Remove parentheses.
52(4(2+n))2=116\frac{{5}^{2}}{{(4(2+n))}^{2}}=\frac{1}{16}

2
Simplify  52{5}^{2}  to  2525.
25(4(2+n))2=116\frac{25}{{(4(2+n))}^{2}}=\frac{1}{16}

3
Use Multiplication Distributive Property: (xy)a=xaya{(xy)}^{a}={x}^{a}{y}^{a}.
2542(2+n)2=116\frac{25}{{4}^{2}{(2+n)}^{2}}=\frac{1}{16}

4
Simplify  42{4}^{2}  to  1616.
2516(2+n)2=116\frac{25}{16{(2+n)}^{2}}=\frac{1}{16}

5
Multiply both sides by 16(2+n)216{(2+n)}^{2}.
25=116×16(2+n)225=\frac{1}{16}\times 16{(2+n)}^{2}

6
Cancel 1616.
25=(2+n)225={(2+n)}^{2}

7
Take the square root of both sides.
±25=2+n\pm \sqrt{25}=2+n

8
Since 5×5=255\times 5=25, the square root of 2525 is 55.
±5=2+n\pm 5=2+n

9
Switch sides.
2+n=±52+n=\pm 5

10
Break down the problem into these 2 equations.
2+n=52+n=5
2+n=−52+n=-5

11
Solve the 1st equation: 2+n=52+n=5.
n=3n=3

12
Solve the 2nd equation: 2+n=−52+n=-5.
n=−7n=-7

13
Collect all solutions.
n=3,−7n=3,-7

Done
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