Question

How do you multiply and simplify `(11b^7)/(7q^4) * (49q^6)/(121b)`?

Answer 1

`(7b^6q^2)/11`

Explanation:

As a general approach for simplifying algebraic fractions like this

• Work with the negative signs (does not apply here)
• work with numbers
• work with the indices

`(11b^7)/(7q^4) xx (49q^6)/(121b)" "larr` simplify the numbers first

=`(cancel11b^7)/(cancel7q^4) xx (cancel49^7q^6)/(cancel121^11b)`

Simplify into one numerator and one denominator

=`(7b^7q^6)/(11bq^4)" "larr` subtract the indices of like bases

=`(7b^6q^2)/11`

Answer 2

`(7b^6q^2)/11`

Explanation:

`color(green)("A note about the method")`

Consider the example of: `" "color(blue)(4/5)xxcolor(red)(10/8)`

As this is multiply you can swap things round. The proper name for this is that it has the property of being 'commutative'

so we can write it as: `(color(blue)(4))/(color(red)(8))xx(color(red)(10))/(color(blue)(5))` and still get the same answer.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given:`" "(11b^7)/(7q^4)xx(49q^6)/(121b)`..........Expression (1)

Write as:`" "(11b^7)/(121b)xx(49q^6)/(7q^4)`

`color(brown)("Dealing just with the numbers first")`

`" "(cancel(11)^1b^7)/(cancel(121)^11b)xx(cancel(49)^7q^6)/(cancel(7)^1q^4)`

So now we have:

`b^7/(11b)xx(7q^6)/q^4 ............... color(green)("Expression(2)")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Dealing just with the letters (variables)")`

Consider:`" "b^7/b` this is the same as `(cancel(b)^1xxb^6)/(cancel(b)^1) = b^6`

Consider:`" "q^6/q^4` this is the same as `(q^4xxq^2)/q^4=q^2`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Putting it all together")`

Substituting the variables (letters) into `color(green)("Expression(2)")` we have

`b^6/11xx7q^2...............".Expression "(2_a)`

Giving:`" "(7b^6q^2)/11`