How do you simplify $(- 2 a^{7}) (5 a^{- 2})$ $(-2a^7)(5a^-2)$ ?

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How do you simplify $(- 2 a^{7}) (5 a^{- 2})$ $(-2a^7)(5a^-2)$ ?

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Answer 1 · 2016-05-04T09:29:56

$- 10 a^{5}$ $-10a^5$

Explanation:

$How you would normally see the calculation$ $color(red)("How you would normally see the calculation")$

variants on:

$- (2 \times 5) (a^{7 - 2}) = - 10 a^{5}$ $color(red)(-(2xx5)(a^(7-2)) = -10a^5)$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$Very detailed solution split into 3 parts$ $color(green)("Very detailed solution split into 3 parts")$

Part 1 $\to$ $->$ signs
Part 2 $\to$ $->$ numbers
Part 3 $\to$ $->$ the letters (variables)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Part 1 \to signs$ $color(blue)("Part 1 "-> " signs")$

Write the signs as -1 and +1 $" "$ ( this does work!)

The signs are not the same as each other so

$(- 1) \times (+ 1) = (- 1)$ $(-1)xx(+1) = color(red)((-1))$

$So our answer is a negative value$ $color(blue)("So our answer is a negative value ")$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Part 2 \to numbers$ $color(blue)("Part 2 "-> " numbers")$

$2 \times 5 = 10$ $2xx5 =color(red)(10)$
$The number part of the answer is 10$ $color(blue)("The number part of the answer is 10")$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Part 3 \to letters (variables)$ $color(blue)("Part 3 "-> " letters (variables)")$

We have: $a^{7} \times a^{- 2}$ $" "a^7xxa^(-2)$

$Shortcut method$ $color(brown)("Shortcut method")$
Because the letters (variables) are the same we can do this:

$a^{7} \times a^{- 2} = a^{7 - 2} = a^{5}$ $a^7xxa^(-2)" "=" "a^(7-2)" "=color(red)(a^5)$
'......................................................

$First principles method$ $color(brown)("First principles method")$

$a^{- 2} is the same thing as \frac{1}{a^{2}}$ $a^(-2)" is the same thing as "1/a^2$

So:: $a^{7} \times a^{- 2} = a^{7} \times \frac{1}{a^{2}}$ $" "a^7xxa^(-2)" " =" " a^7xx1/a^2$

Let $a^{7}$ $a^7$ be written as $a^{5} \times a^{2}$ $a^5xxa^2$

Then we have: $a^{5} \times a^{2} \times \frac{1}{a^{2}}$ $a^5xxa^2xx1/a^2$

This is the same as
$a^{5} \times \frac{a^{2}}{a^{2}}$ $" "a^5xxa^2/a^2$

But $\frac{a^{2}}{a^{2}} = 1$ $a^2/a^2=1$

$a^{5} \times \frac{a^{2}}{a^{2}} = a^{5}$ $" "a^5xxa^2/a^2" "=" "color(red)(a^5)$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$putting it all together$ $color(blue)("putting it all together")$

$(- 2 a^{7}) (5 a^{- 2}) = - 10 a^{5}$ $color(red)((-2a^7)(5a^(-2))= -10a^5)$

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How do you simplify $(- 2 a^{7}) (5 a^{- 2})$ $(-2a^7)(5a^-2)$ ?

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