# If x = 4, what is x^-3/2 ( x^100/ x^99)?

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Let f(x) = x^-3/2 (x^100/x^99)

First we need to simplify the function:

We know that: x^a/x^b = x^(a-b)

==> f(x) = x^-3/2 (x^100-99)

= x^-3/2 ( x)

Also we know that:

x^-a/b = 1/x^a/b

==> f(x) = x/x^3/2

= x^(1-3/2)

= x^-1/2

= 1/x^1/2

==> f(x) = 1/sqrtx

Now given x= 4

==> f(4) = 1sqrt4= 1/2

**==> f(4) = 1/2**

We'll start with the term x^-3/2.

We'll apply the rule of the negative power:

x^-n = 1/x^n

Now, we'll substitute n by -3/2:

x^-3/2 = 1/x^3/2

Now, we'll compute the ratio x^100/x^99.

Since the bases x are matching, we'll subtract the exponents:

x^100/x^99 = x^(100-99) = x^1 = x

We'll re-write the expression:

x^-3/2(x^100/x^99) = x^(100-99)/x^3/2 = x^1/x^3/2

x^-3/2(x^100/x^99) = x^(1 - 3/2)

x^-3/2(x^100/x^99) = x^-1/2

x^-3/2(x^100/x^99) = 1/x^1/2

x^-3/2(x^100/x^99) = 1/sqrt x

**x^-3/2(x^100/x^99) = (sqrt x)/x**

We'll substitute x by 4:

(sqrt x)/x = sqrt4/4

sqrt4/4 = 2/4

**sqrt4/4 = 1/2**