Please help me to understand

[nautudent]

I have recently stumbled across this following question:

Given that x = 2 ^p and y = 2 ^s, express the following in terms of x and y:

i) 2 ^p+s
ii) 2 ^2p
iii) 2 ^p-1

The answers are the following:

Answers

i) xy
ii) x²
iii) (1/2) x

I would like to know how to work out the answers as i got them wrong.
i)I understand why it would be xy but i dont understand why if it is xy the number would be 2^p+s and not 4^p+s
ii)I completely don't understand this
iii)Completely confused with this one as well


It would be great if someone could help and say how those answers came about
Thanks a lot

 

[Otis]

Hi. Here's a basic property of exponents.

2^p * 2^s = 2^(p+s)

In other words, if two powers with the same base are multiplied, then the exponents are added.

This works in reverse, too. That is, if you're given a power where the exponent is a sum, you may factor the power as a product of two powers with the same base.

2^(p + s) = 2^p * 2^s

Simple substitutions for 2^p and 2^s yield x*y. The 2s are not multiplied because that's not what the property says.

In exercise iii, realize that the exponent p-1 is the same as p + (-1). Now use the property above to finish.

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Exercise ii requires a different property of exponents.

(2^p)^2 = 2^(2*p)

In other words, if a power is itself raised to another power, then the exponents are multiplied.

This works in reverse, too. That is, if you're given a power where the exponent is a product, you may rewrite the expression as one power raised to the other exponent.

2^(2*p) = (2^p)^2 or (2^2)^p

Again, a simple substitution in the blue expression yields x^2.

Questions? :cool:

 

[Mrspi]

I have recently stumbled across this following question:

Given that x = 2 ^p and y = 2 ^s, express the following in terms of x and y:

i) 2 ^p+s
ii) 2 ^2p
iii) 2 ^p-1

The answers are the following:

Answers

i) xy
ii) x²
iii) (1/2) x

I would like to know how to work out the answers as i got them wrong.
i)I understand why it would be xy but i dont understand why if it is xy the number would be 2^p+s and not 4^p+s
ii)I completely don't understand this
iii)Completely confused with this one as well

It would be great if someone could help and say how those answers came about
Thanks a lot

These all have to do with understanding the rules of exponents.....

a^b * a^c = a^{b + c}-......to multiply powers of the same base, ADD the exponents
So,
2^{p + s} must have come from 2^p * 2^s
since x = 2^p and y = 2^s, 2^p * 2^s can be written as x*y, or xy.

Another rule:
(a^b)^c = a^b*c......to raise a power to a power, you MULTIPLY the exponents
See if that rule can help you with the second problem.

And a third rule:

(a^b) / (a^c) = a^{b - c}.......to divide two powers of the same base, subtract the exponents. So,
2^{p - 1} must have come from the division of two powers of 2. We know that 2^p = x. What would 2^-1 be?

I hope this helps you.

 

[Otis]

... a third rule:

(a^b) / (a^c) = a^{b - c}.......to divide two powers of the same base, subtract the exponents. So,
2^{p - 1} must have come from the division of two powers of 2. We know that 2^p = x. What would 2^-1 be?

Mrspi! So nice to see you here. Of course. That's likely the intended lesson above.

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Hey nautudent! There are are other rules, too, and one in particular is needed to understand Mrspi's question: What would 2^(-1) be?

Have you seen the property which deals with negative exponents?

Cheers ~ Mark

PS: I'm currently outside of Flag

 

[Mrspi]

Mrspi! So nice to see you here. Of course. That's likely the intended lesson above.

- - - - - - - - - - - - - - - -

Hey nautudent! There are are other rules, too, and one in particular is needed to understand Mrspi's question: What would 2^(-1) be?

Have you seen the property which deals with negative exponents?

Cheers ~ Mark

PS: I'm currently outside of Flag

Hiya, Mark! Good to see you, too. I wondered what had become of you (hard to keep up with all of your different names!)

MrsPi (Tchrjan in another realm.....)