Question

What is `2a^3-:3a^2*6a^5`?

Answer 1

`a^-4/9 = 1/( 9a^4)`

Explanation:

`2xxa^3` = 2 x a x a x a

`3xxa^2` = 3 x a x a

`6xxa^5`= 2 x 3 x a x a x a x a x a

putting these together as a fraction ratio gives

`( 2 xxa xx a xx a ) /( 2 xx 3 xx 3 xx a xx a xx a xx a xx a xx a xx a`

`( cancel2 xx cancel(a xx a xx a) ) /( cancel2 xx 3 xx 3 xx cancel(a xx a xx a) xx a xx a xx a xx a`

This leaves 3 x3 x a x a x a x a on the bottom or

`1/( 9a^4)`

dividing by `a^4` is the same as multiplying by `a^-4`

so `a^-4/9,` but with a positive index is a better form.

Answer 2

It depends. It could be `1/(9a^4)` or `4a^6`

Explanation:

The expression `2a^3 -: 3a^2*6a^5` is ambiguous and can be interpreted in at least two ways depending on convention:

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Interpretation 1

`2a^3 -: 3a^2*6a^5 = (2a^3) / (3a^2*6a^5)`

We arrive at this interpretation for one/both of the following reasons:

• The obelus `-:` is taken to mean that the whole expression on the left should be divide by the whole expression on the right.

• Multiplication is understood to have higher precendence than division.

Accordingly, we find:

`2a^3 -: 3a^2*6a^5 = (2a^3) / (3a^2*6a^5)=(2a^3)/(18a^7)=1/(9a^4)=1/9a^(-4)`

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Interpretation 2

`2a^3 -: 3a^2*6a^5 = (2a^3)/(3a^2)*6a^5`

We may arrive at this interpretation for the following reason:

• Guided by PEMDAS, BODMAS or BIDMAS, we consider division and multiplication to have equal precedence, so should be evaluated left to right. Note that for this understanding we consider the multiplication by juxtaposition in each of the expressions `2a^3`, `3a^2` and `6a^5` as of higher precedence - so it is not pure PEMDAS, etc.

Accordingly we find:

`2a^3 -: 3a^2*6a^5 = (2a^3)/(3a^2)*6a^5=2/3a*6a^5 = 4a^6`

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Remarks

Operator precence conventions are supposed to help resolve ambiguities like this, but if the writer and reader of an expression may have different conventions the intention can be misunderstood. It would be better if parentheses were used to make the intended meaning clear, or the obelus `-:` avoided.