Solution

Since both quantities are algebraic expressions, another way to approach the comparison is to set up a placeholder relationship, denoted by |?|, between the two quantities and then to simplify to see what conclusions you can draw. As you simplify and draw conclusions, keeping in mind that x is a negative integer.

\(2^x vs 3^x^+^1\)

or \(2^x vs 3* 3^x\)
or \(\frac{2^x}{3^x} vs 3\)

Let n = -x and since x is negative integer n is a positive integer.

or \((\frac{2}{3})^{-n} vs 3\)

or \((\frac{3}{2})^n vs 3\)

Now clearly for some values of n Quantity A is bigger (for example n = 1) and other values Quantity B is greater (for example n=3). Hence a relation ship between the two cannot be determined. Option D is correct.
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Let's plug in some values for x and see what happens

x = -1
We get:
Quantity A: 2^(-1) = 1/2
Quantity B: 3^(-1 + 1) = 3^0 = 1
Quantity B is greater

x = -3
We get:
Quantity A: 2^(-3) = 1/8
Quantity B: 3^(-3 + 1) = 3^(-2) = 1/9
Quantity A is greater

Answer:

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Can someone please tell me why 3^x+1 is the same as 3∗3^x?

Or can just tell me the name of the rule?

I'm sure I should know it, but I don't.

Thanks for your help.

It's called the Product Law, which says (b^x)(b^y) = b^(x + y)
For example, (2^5)(2^3) = 2^(5 + 3) = 2^8

So, (3)(3^x) = (3^1)(3^x)
= 3^(1 + x)
= 3^(x + 1)

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