The value of `27^(-1//3) xx [27^(2//3) + 27^(1//3)]` is

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the expression \( 27^{-\frac{1}{3}} \times [27^{\frac{2}{3}} + 27^{\frac{1}{3}}] \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 27^{-\frac{1}{3}} \times [27^{\frac{2}{3}} + 27^{\frac{1}{3}}] \]

### Step 2: Apply the property of indices
Using the property of indices \( a^m \times a^n = a^{m+n} \), we can distribute \( 27^{-\frac{1}{3}} \) to both terms inside the brackets:
\[ = 27^{-\frac{1}{3}} \times 27^{\frac{2}{3}} + 27^{-\frac{1}{3}} \times 27^{\frac{1}{3}} \]

### Step 3: Simplify each term
Now, we simplify each term using the property of indices:
1. For the first term:
\[ 27^{-\frac{1}{3}} \times 27^{\frac{2}{3}} = 27^{-\frac{1}{3} + \frac{2}{3}} = 27^{\frac{1}{3}} \]

2. For the second term:
\[ 27^{-\frac{1}{3}} \times 27^{\frac{1}{3}} = 27^{-\frac{1}{3} + \frac{1}{3}} = 27^{0} = 1 \]

### Step 4: Combine the results
Now, we can combine the results from the two terms:
\[ 27^{\frac{1}{3}} + 1 \]

### Step 5: Evaluate \( 27^{\frac{1}{3}} \)
We know that:
\[ 27 = 3^3 \implies 27^{\frac{1}{3}} = (3^3)^{\frac{1}{3}} = 3^{3 \times \frac{1}{3}} = 3^1 = 3 \]

### Step 6: Final result
Now substituting back, we have:
\[ 3 + 1 = 4 \]

Thus, the value of the expression \( 27^{-\frac{1}{3}} \times [27^{\frac{2}{3}} + 27^{\frac{1}{3}}] \) is \( \boxed{4} \).

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