To simplify the expression $\left(\frac{x}{16}\right)^{-\frac{3}{4}}$, follow these steps:

Step 1: Apply the Negative Exponent Rule

A negative exponent means you take the reciprocal of the base. The rule is: $a^{-n} = \frac{1}{a^n}$ Thus, the expression becomes: $\left(\frac{x}{16}\right)^{-\frac{3}{4}} = \frac{1}{\left(\frac{x}{16}\right)^{\frac{3}{4}}}$

Step 2: Apply the Fractional Exponent

Next, distribute the fractional exponent $\frac{3}{4}$ to both $x$ and $16$: $\left(\frac{x}{16}\right)^{\frac{3}{4}} = \frac{x^{\frac{3}{4}}}{16^{\frac{3}{4}}}$

Step 3: Simplify $16^{\frac{3}{4}}$

Since $16 = 2^4$, you can simplify: $16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^3 = 8$

So the expression becomes: $\frac{x^{\frac{3}{4}}}{8}$

Step 4: Combine the Results

Now, put everything together: $\frac{1}{\left(\frac{x^{\frac{3}{4}}}{8}\right)} = \frac{8}{x^{\frac{3}{4}}}$

Thus, the simplified expression is: $\frac{8}{x^{\frac{3}{4}}}$

Would you like further clarification on any step?

5 Related Questions:

1. How do you simplify expressions with negative exponents?
2. What are the rules for handling fractional exponents?
3. How do you simplify expressions involving powers of fractions?
4. Can you provide an example where the base of the fraction is a variable instead of a number?
5. How can we apply the exponent rules to more complex expressions?

Tip: When simplifying expressions with fractional exponents, always simplify powers of numbers before handling the variables.