Simplify `6sqrt((27)^(-2/3)+(8)^(-2/3))`

Options

A. `root(6)(35)`
B. `6/sqrt13`
C. `sqrt13`
D. `root(6)(6)`

Answer

To simplify the expression \( 6\sqrt{(27)^{-2/3} + (8)^{-2/3}} \), we can follow these steps:

### Step 1: Rewrite the expression using powers
We start with the expression:
\[ 6\sqrt{(27)^{-2/3} + (8)^{-2/3}} \]
We can express \( 27 \) and \( 8 \) as powers:
\[ 27 = 3^3 \quad \text{and} \quad 8 = 2^3 \]
Thus, we rewrite the expression:
\[ 6\sqrt{(3^3)^{-2/3} + (2^3)^{-2/3}} \]

### Step 2: Apply the power of a power rule
Using the power of a power rule \( (a^m)^n = a^{m \cdot n} \), we simplify:
\[ (3^3)^{-2/3} = 3^{3 \cdot (-2/3)} = 3^{-2} \quad \text{and} \quad (2^3)^{-2/3} = 2^{3 \cdot (-2/3)} = 2^{-2} \]
So we can rewrite the expression as:
\[ 6\sqrt{3^{-2} + 2^{-2}} \]

### Step 3: Rewrite negative exponents
Next, we rewrite the negative exponents:
\[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \quad \text{and} \quad 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \]
Thus, we have:
\[ 6\sqrt{\frac{1}{9} + \frac{1}{4}} \]

### Step 4: Find a common denominator
To add the fractions \( \frac{1}{9} + \frac{1}{4} \), we need a common denominator. The least common multiple of \( 9 \) and \( 4 \) is \( 36 \):
\[ \frac{1}{9} = \frac{4}{36} \quad \text{and} \quad \frac{1}{4} = \frac{9}{36} \]
Thus:
\[ \frac{1}{9} + \frac{1}{4} = \frac{4}{36} + \frac{9}{36} = \frac{13}{36} \]

### Step 5: Substitute back into the expression
Now we substitute back into the square root:
\[ 6\sqrt{\frac{13}{36}} \]

### Step 6: Simplify the square root
We can simplify the square root:
\[ \sqrt{\frac{13}{36}} = \frac{\sqrt{13}}{\sqrt{36}} = \frac{\sqrt{13}}{6} \]
Thus, we have:
\[ 6 \cdot \frac{\sqrt{13}}{6} = \sqrt{13} \]

### Final Answer
The simplified expression is:
\[ \sqrt{13} \]