Problem of the Week

Updated at Dec 18, 2023 9:20 AM

This week's problem comes from the equation category.

How would you solve 6(4−4m)2=124\frac{6}{{(4-4m)}^{2}}=\frac{1}{24}?

Let's begin!



6(4−4m)2=124\frac{6}{{(4-4m)}^{2}}=\frac{1}{24}

1
Factor out the common term 44.
6(4(1−m))2=124\frac{6}{{(4(1-m))}^{2}}=\frac{1}{24}

2
Use Multiplication Distributive Property: (xy)a=xaya{(xy)}^{a}={x}^{a}{y}^{a}.
642(1−m)2=124\frac{6}{{4}^{2}{(1-m)}^{2}}=\frac{1}{24}

3
Simplify  42{4}^{2}  to  1616.
616(1−m)2=124\frac{6}{16{(1-m)}^{2}}=\frac{1}{24}

4
Simplify  616(1−m)2\frac{6}{16{(1-m)}^{2}}  to  38(1−m)2\frac{3}{8{(1-m)}^{2}}.
38(1−m)2=124\frac{3}{8{(1-m)}^{2}}=\frac{1}{24}

5
Multiply both sides by 8(1−m)28{(1-m)}^{2}.
3=124×8(1−m)23=\frac{1}{24}\times 8{(1-m)}^{2}

6
Use this rule: ab×cd=acbd\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}.
3=1×8(1−m)2243=\frac{1\times 8{(1-m)}^{2}}{24}

7
Simplify  1×8(1−m)21\times 8{(1-m)}^{2}  to  8(1−m)28{(1-m)}^{2}.
3=8(1−m)2243=\frac{8{(1-m)}^{2}}{24}

8
Simplify  8(1−m)224\frac{8{(1-m)}^{2}}{24}  to  (1−m)23\frac{{(1-m)}^{2}}{3}.
3=(1−m)233=\frac{{(1-m)}^{2}}{3}

9
Multiply both sides by 33.
3×3=(1−m)23\times 3={(1-m)}^{2}

10
Simplify  3×33\times 3  to  99.
9=(1−m)29={(1-m)}^{2}

11
Take the square root of both sides.
±9=1−m\pm \sqrt{9}=1-m

12
Since 3×3=93\times 3=9, the square root of 99 is 33.
±3=1−m\pm 3=1-m

13
Switch sides.
1−m=±31-m=\pm 3

14
Break down the problem into these 2 equations.
1−m=31-m=3
1−m=−31-m=-3

15
Solve the 1st equation: 1−m=31-m=3.
m=−2m=-2

16
Solve the 2nd equation: 1−m=−31-m=-3.
m=4m=4

17
Collect all solutions.
m=−2,4m=-2,4

Done
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