4/(216 ^ (-3/2)) + 1/((256) ^ (-3/4)) + 7/((243) ^ (-1/5))

Let's simplify the expression step by step:

\[
\frac{4}{216^{-\frac{3}{2}}} + \frac{1}{256^{-\frac{3}{4}}} + \frac{7}{243^{-\frac{1}{5}}}
\]

### Step 1: Apply the negative exponents
Recall that \(a^{-n} = \frac{1}{a^n}\). So, the negative exponents in the denominator will flip the base to the numerator.

- \( 216^{-\frac{3}{2}} = \frac{1}{216^{\frac{3}{2}}} \) → we can rewrite this as \( \frac{1}{216^{-\frac{3}{2}}} = 216^{\frac{3}{2}} \).
- \( 256^{-\frac{3}{4}} = 256^{\frac{3}{4}} \).
- \( 243^{-\frac{1}{5}} = 243^{\frac{1}{5}} \).

Thus, the expression becomes:

\[
4 \cdot 216^{\frac{3}{2}} + 1 \cdot 256^{\frac{3}{4}} + 7 \cdot 243^{\frac{1}{5}}
\]

### Step 2: Evaluate the exponents
Now calculate the values of the terms individually.

- \( 216^{\frac{3}{2}} = (6^3)^{\frac{1}{2}} = 6^3 = 6 \times 6 \times 6 = 216 \).
- \( 256^{\frac{3}{4}} = (4^4)^{\frac{3}{4}} = 4^3 = 64 \).
- \( 243^{\frac{1}{5}} = 3^5 = 243 \), so \( 243^{\frac{1}{5}} = 3 \).

### Step 3: Multiply and add the values
Now substitute these values back into the expression:

\[
4 \times 216 + 1 \times 64 + 7 \times 3 = 864 + 64 + 21 = 949
\]

### Final Answer:
\[
\boxed{949}
\]

Would you like any further details or clarifications?

Here are some related questions to explore:

1. How do fractional exponents relate to roots?
2. What are the steps for simplifying an expression with both exponents and negative bases?
3. How do you handle expressions with different bases and exponents?
4. Why does a negative exponent flip the fraction?
5. Can you apply similar rules for simplifying radical expressions?

**Tip:** When simplifying expressions with exponents, it helps to break them down into prime factors and use exponent rules carefully.

Learn how to simplify the expression 4/(216^(-3/2)) + 1/(256^(-3/4)) + 7/(243^(-1/5)) step by step using exponent rules. This guide covers the handling of negative and fractional exponents, suitable for students in Grades 9-11.

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