Format for answering equations - exponents and roots

[BilalH766]

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I want to know why certain numbers are left in their exponential form and others are not. Like in this image. Why do we put 16 for a 2 to the power of 4 in one situation but not in the other?

[RhettA356]

My guess: so we can easily eliminate the 2^5 from the denominator (bottom) by changing 2^12 to 2^7 in the numerator (top). The ^5 cancel each other out, so it's easier to just leave it as 2^5 on the bottom instead of changing it to 16, then changing it back to 2^5 in order to cancel.

[RonPurewal]

you have to look FUNCTIONALLY at what's happening here. it depends on what you're going to DO with those quantities.

• with the 2^4 that's inside the parentheses, you're trying to SUBTRACT something from that.
for addition/subtraction, exponent form is totally useless—this is why quadratics look like quadratics, and can't be collapsed into a single term, for instance—so, you take 2^4 out of exponent form, and make it 16.
then you can easily subtract 16 – 1.

• for the 2^4 that's on the bottom, you're trying to do DIVISION.
this is just the familiar rule where you subtract the exponents. no need to take anything out of exponent form.

[RonPurewal]

more generally, this is a microcosm of the way you have to treat the whole exam—you have to be REACTIVE.

you have to pay attention to what's actually happening in each particular situation, and make decisions accordingly. you can't hope to memorize some sort of universal protocol.

[BilalH766]

ok thanks everyone. I think I understand.

[RonPurewal]

you're welcome.