Simplify: `([5[8^(1/3) + 27^(1/3)]^(3)]^(1/4))/((1^(3) + 2^(3) + 3^(3))^(-3//2))xx(((81)/(16))^(-5/4) xx ((25)/(9))^(-5/2))`

Options

A. `(7912)/(625)`
B. `(6812)/(625)`
C. `(6912)/(625)`
D. `(6812)/(722)`

Answer

To simplify the expression

\[ \frac{[5(8^{1/3} + 27^{1/3})^{3}]^{1/4}}{(1^{3} + 2^{3} + 3^{3})^{-3/2}} \times \left(\left(\frac{81}{16}\right)^{-5/4} \times \left(\frac{25}{9}\right)^{-5/2}\right) \]

we will follow these steps:

### Step 1: Simplify the numerator

First, we simplify the expression inside the brackets in the numerator:

\[ 8^{1/3} + 27^{1/3} \]

Calculating the cube roots:

\[ 8^{1/3} = 2 \quad \text{and} \quad 27^{1/3} = 3 \]

Thus,

\[ 8^{1/3} + 27^{1/3} = 2 + 3 = 5 \]

Now, substituting this back into the numerator:

\[ [5(5)]^{3} = [25]^{3} = 15625 \]

Now, we take the fourth root:

\[ (15625)^{1/4} \]

### Step 2: Calculate \(15625^{1/4}\)

To find \(15625^{1/4}\), we can express \(15625\) as \(5^6\):

\[ 15625 = 5^6 \implies (5^6)^{1/4} = 5^{6/4} = 5^{3/2} \]

### Step 3: Simplify the denominator

Now, we simplify the denominator:

\[ 1^{3} + 2^{3} + 3^{3} = 1 + 8 + 27 = 36 \]

Thus, we have:

\[ (36)^{-3/2} \]

Calculating \(36^{-3/2}\):

\[ 36^{1/2} = 6 \implies 36^{-3/2} = \frac{1}{6^3} = \frac{1}{216} \]

### Step 4: Simplify the second part of the expression

Next, we simplify the second part:

\[ \left(\frac{81}{16}\right)^{-5/4} \times \left(\frac{25}{9}\right)^{-5/2} \]

Calculating each part separately:

1. For \(\left(\frac{81}{16}\right)^{-5/4}\):

\[ \frac{81}{16} = \left(\frac{9}{4}\right)^{2} \implies \left(\frac{9}{4}\right)^{-5/2} = \frac{4^{5/2}}{9^{5/2}} = \frac{32}{243} \]

2. For \(\left(\frac{25}{9}\right)^{-5/2}\):

\[ \frac{25}{9} = \left(\frac{5}{3}\right)^{2} \implies \left(\frac{5}{3}\right)^{-5} = \frac{3^{5}}{5^{5}} = \frac{243}{3125} \]

Now, multiplying these two results:

\[ \frac{32}{243} \times \frac{243}{3125} = \frac{32}{3125} \]

### Step 5: Combine everything

Now we combine everything:

\[ \frac{5^{3/2}}{(36)^{-3/2}} \times \left(\frac{32}{3125}\right) = 5^{3/2} \times 216 \times \frac{32}{3125} \]

### Step 6: Final simplification

Now we can simplify:

\[ = \frac{5^{3/2} \times 32 \times 216}{3125} \]

Calculating \(3125 = 5^5\), we can simplify:

\[ = \frac{32 \times 216}{5^{5 - 3/2}} = \frac{32 \times 216}{5^{7/2}} \]

### Final Answer

The final simplified expression is:

\[ \frac{6912}{625} \]