Question

Evaluate the function

can someone please explain how I would solve this please?

Answer 1

To think intuitively of what it means to take a number to a negative power, first consider how we define taking a number to a positive power.

Before we even get there, though, consider how we define multiplication as repeated addition. When you see an expression like 3 x 5, what that essentially translates to is "add 3 to itself 5 times," so we could also write 3 x 5 as 3 + 3 + 3 + 3 + 3. Having established that, around middle school, you'll typically get your first exposure to positive exponents, which are defined at first as repeated multiplication. When you see something like
`3^5`, we could also read that as "multiply 3 by itself 5 times," or
`3*3*3*3*3`.

With that definition for positive exponents defined, it makes sense that we would define negative exponents in terms of the inverse of repeated multiplication: repeated division. Each time we step the exponent back by 1, we divide by the base again. For example, let's take these decreasing powers of 3 and notice what happens:


`3^3 = 27\\3^2=27/3=9\\3^1=9/3=3\\3^0=3/3=1`

If we start stepping back further, we start getting to values below 1:


`3^(-1)=1/3\\3^(-2)=(1/3)/3=1/3^2=1/9\\ 3^(-3)=(1/9)/3=1/3^3=1/27`

And this pattern continues, with the essential takeaway being that
`3^(-x)=1/3^x`

Try applying that pattern to the equation you've been given,
`y=4\cdot2^(-6)`, and see what you get!

Answer 2

y = 4 x 2^-6
y = 4 x 1/2^6
y = 4 x 1/64
y = 1/16