Question

How do you simplify `\frac { - 49x ^ { 8} + 14x ^ { 6} - 14x ^ { 4} } { - 7x ^ { 6} }`?

Answer 1

`=> 7x ^2 - 2 x^-2+ 2`

OR

`=> 7x ^2 - 2/ x^2+ 2`

Explanation:

`\frac { - 49x ^ { 8} + 14x ^ { 6} - 14x ^ { 4} } { - 7x ^ { 6} }`

Take `7x^4` common from both numerator and denominator:

Using Rule of indices :
`a^m a^n = a^(m+n)`

=`\frac {7x^4 (- 7x ^ { 4} + 2x ^ { 2} - 2)} { - 7x ^ { 4} (x^2) }`

=`\frac {cancel(7x^4) (- 7x ^ { 4} + 2x ^ { 2} - 2)} { - cancel(7x ^ { 4}) (x^2) }`

=`\frac { (- 7x ^ { 4} + 2x ^ { 2} - 2)} { - (x^2) }`

=`\frac {cancel- (7x ^ { 4} - 2x ^ { 2} + 2)} {cancel- (x^2) }`

=`\frac { (7x ^ { 4}- 2x ^ { 2} + 2)} { (x^2) }`

Using rule of indices: `a^m /a^n = a^(m-n)` and/or `1/a^m = a^-m`

=`(7x ^ { 4} + 2x ^ { 2} - 2)\times (x^-2)`

=`7x ^ { 4-2} + 2x ^ { 2-2} - 2 x^-2`

=`7x ^ { 2} + 2x ^ { 0} - 2 x^-2`

`x^0 = 1`

=`7x ^2 + 2\times 1 - 2 x^-2`

=`7x ^2 + 2 - 2 x^-2`

=`7x ^2 - 2 x^-2+ 2`

OR

=`7x ^2 - 2/ x^2+ 2`

Answer 2

`color(magenta)(7x^2-2+2x^-2`

Explanation:

`(-49x^8+14x^6-14x^4)/(-7x^6)`

`color(white)(..........)ulcolor(white)(......)ul(7x^2-2+2x^-2)`
`-7x^6|-49x^8+14x^6-14x^4`
`color(white)(.............)ul(-49x^8)`
`color(white)(............................)14x^6`
`color(white)(............................)ul(14x^6)`
`color(white)(...................................)-14x^4`
`color(white)(.....................................)ul(-14x^4)`

`color(magenta)((-49x^8+14x^6-14x^4)/(-7x^6) = 7x^2-2+2x^-2`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Check:-

`color(white)(.............)7x^2-2+2x^-2`
`xxcolor(white)(....)-7x^6`
`color(white)(.........)overline(-49x^8+14x^6-14x^4)`