Question

How do you evaluate `(- \frac { 7} { 5} ) ^ { 2} - 2( \frac { 1} { 20} - \frac { 1} { 4} )`?

Answer 1

`59/25`

Explanation:

First, you need to square `-7/5`:

`(-7/5)^2 = (-7/5)(-7/5)=49/25`

Now, you need to combine the fractions on the far right side:

`−2(1/20−1/4)=(-1/10+1/2)`

Find common denominators (`10`):

`(-1/10+1/2)=(-1/10+(1(5))/(2(5)))=(-1/10+5/10)`

Simplify:

`(-1/10+5/10)=(4/10)`

Combine `49/25` and `4/10` to get the simplified version:

`(49(2))/(25(2))+(4(5))/(10(5))=118/50=59/25`

Answer 2

`2 9/25 color(white)("dd") ->color(white)("dd") 59/25`

Explanation:

`color(magenta)("There is a trap about signs in this - easy to forget about something")`

`color(blue)("Consider "(-7/5)^2`

First step:

When multiplying or dividing like signs give positive and unlick signs give negative.

We have a negative multiplied by a negative. So the answer to this part is positive `(-7/2)^2->+" sum number"`

Second step:

Dealing with just the numbers: `7^2/5^2=49/25`

Looking ahead it would work better if the denominator is 100.

Multiply by 1 and you do not change the value. However 1 comes in many forms.

`color(purple)(+49/25color(red)(xx1)color(white)("dd") ->color(white)("dd")+49/25color(red)(xx4/4)color(white)("dd") =color(white)("dd") +196/100)`

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`color(blue)("Consider "-2(1/20-1/4)`

Multiply everything inside the bracket by `-2` giving

`-2/20+2/4 color(white)("d")->color(white)("d")2/4-2/20`

Because of what we had for the first part I am choosing to make the denominators 100

`color(green)(color(white)("ddddddddd")->color(white)("d") [2/4color(red)(xx1)]-[2/20color(red)(xx1)]`

`color(green)(color(white)("ddddddddd")-> [2/4color(red)(xx25/25)]-[2/20color(red)(xx5/5)]`

`color(purple)(color(white)("ddddddddd")->color(white)("ddd")50/100color(white)("dd")-color(white)("dd")10/100color(white)("ddd") = +40/100)`
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`color(blue)("Putting "ul("everything")" back together")`

`color(magenta)("Avoiding the trap - note that 'put with' is the same as +")`

`(-7/5)^2 color(white)("d")"put with"color(white)("d")[-2(1/20-1/4)]`

`color(white)("dd")196/100 color(white)("d")" put with"color(white)("ddd")[+40/100]`

`color(purple)(color(white)("dddddd")196/100+40/100color(white)("ddddd") = color(white)("ddd")236/100color(white)("ddd")=color(white)("d")2 9/25color(white)("ddd") -> 59/25)`