Problem of the Week

Updated at Nov 2, 2020 3:42 PM

How can we solve the equation 5×6(4x)2=3405\times \frac{6}{{(4x)}^{2}}=\frac{3}{40}?

Below is the solution.



5×6(4x)2=3405\times \frac{6}{{(4x)}^{2}}=\frac{3}{40}

1
Use Multiplication Distributive Property: (xy)a=xaya{(xy)}^{a}={x}^{a}{y}^{a}.
5×642x2=3405\times \frac{6}{{4}^{2}{x}^{2}}=\frac{3}{40}

2
Simplify  42{4}^{2}  to  1616.
5×616x2=3405\times \frac{6}{16{x}^{2}}=\frac{3}{40}

3
Simplify  5×616x25\times \frac{6}{16{x}^{2}}  to  3016x2\frac{30}{16{x}^{2}}.
3016x2=340\frac{30}{16{x}^{2}}=\frac{3}{40}

4
Simplify  3016x2\frac{30}{16{x}^{2}}  to  158x2\frac{15}{8{x}^{2}}.
158x2=340\frac{15}{8{x}^{2}}=\frac{3}{40}

5
Multiply both sides by 8x28{x}^{2}.
15=340×8x215=\frac{3}{40}\times 8{x}^{2}

6
Use this rule: ab×cd=acbd\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}.
15=3×8x24015=\frac{3\times 8{x}^{2}}{40}

7
Simplify  3×8x23\times 8{x}^{2}  to  24x224{x}^{2}.
15=24x24015=\frac{24{x}^{2}}{40}

8
Simplify  24x240\frac{24{x}^{2}}{40}  to  3x25\frac{3{x}^{2}}{5}.
15=3x2515=\frac{3{x}^{2}}{5}

9
Multiply both sides by 55.
15×5=3x215\times 5=3{x}^{2}

10
Simplify  15×515\times 5  to  7575.
75=3x275=3{x}^{2}

11
Divide both sides by 33.
753=x2\frac{75}{3}={x}^{2}

12
Simplify  753\frac{75}{3}  to  2525.
25=x225={x}^{2}

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

14
Since 5×5=255\times 5=25, the square root of 2525 is 55.
±5=x\pm 5=x

15
Switch sides.
x=±5x=\pm 5

Done
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