Simplify and write with positive exponet

(4x^-1)^-1

Work:
(1/4x)^-1
(1/(1/4x))(4x/4x)= 1

(X^3+3x^2+3x+1)^-2/3

Work
1/(X^3+3x^2+3x+1)^2/3
1/(X^2+52/25x^4/3+52/25x+1)

I'm confused on whether these are right or not can someone check my work or explain to me why they're wrong if they are?

how about this way:

using (ab)^n = (a^n)(b^n)

(4x^-1)^-1
= (4^-1)(x^-1)^-1
= (1/4)(x^1)
= (1/4)x
or
= x/4

#2

(x^3+3x^2+3x+1)^(-2/3)
= 1/(x^3+3x^2+3x+1)^(2/3)

notice I assumed your exponent was -2/3,
the way you typed it, the exponent would have been -2 , with 3 a divisor
( (x^3+3x^2+3x+1)^-2 )/3

your last line of 1/(X^2+52/25x^4/3+52/25x+1)
makes absolutely no sense to me

Let's go step by step to check your work for each expression.

For the first expression, (4x^-1)^-1:

Step 1: Start with the rule that when raising a term to a negative exponent, it can be moved to the denominator by changing the sign of the exponent.
(4x^-1)^-1 = 1 / (4x^-1)

Step 2: Now, simplify the numerator:
1 numerator means it stays the same.

Step 3: Simplify the denominator:
For x^-1, you can rewrite it as 1/x.
So, 4x^-1 becomes 4 * (1/x) = 4/x.

Combining the numerator and denominator, we have:
1 / (4/x) = x/4

Therefore, the simplified expression is x/4.

Now let's move on to the second expression, (x^3 + 3x^2 + 3x + 1)^-2/3:

Step 1: Apply the rule that when a term is raised to a fractional exponent, it can be re-written as a root.
(x^3 + 3x^2 + 3x + 1)^(-2/3) = (1 / (x^3 + 3x^2 + 3x + 1)^(2/3))

Step 2: Simplify the numerator:
1 in the numerator stays the same.

Step 3: Simplify the denominator:
(x^3 + 3x^2 + 3x + 1) raised to the power of 2/3.

Based on the information given, we cannot simplify this expression any further. The expression you provided is already in its simplest form.

Therefore, the simplified expression is (1 / (x^3 + 3x^2 + 3x + 1)^(2/3)).