`(frac(-2)(3))^(10) div (frac(-2)(3))^(8)=` ?

Options

A. (a) `frac(4)(9)`
B. (b) `frac(-4)(9)`
C. (c) `(frac(-2)(3))^(18)`
D. (d) none of these

Answer

To solve the expression \((\frac{-2}{3})^{10} \div (\frac{-2}{3})^{8}\), we can use the laws of exponents. Here’s a step-by-step solution:

### Step 1: Identify the bases and exponents
The bases in both parts of the expression are the same: \(\frac{-2}{3}\). The exponents are 10 and 8.

### Step 2: Apply the law of exponents for division
According to the laws of exponents, when dividing two expressions with the same base, we subtract the exponents:
\[ a^m \div a^n = a^{m-n} \]
In our case:
\[ (\frac{-2}{3})^{10} \div (\frac{-2}{3})^{8} = (\frac{-2}{3})^{10-8} \]

### Step 3: Perform the subtraction of the exponents
Now we subtract the exponents:
\[ 10 - 8 = 2 \]
So we have:
\[ (\frac{-2}{3})^{10-8} = (\frac{-2}{3})^{2} \]

### Step 4: Calculate the result
Now we can calculate \((\frac{-2}{3})^{2}\):
\[ (\frac{-2}{3})^{2} = \frac{(-2)^{2}}{3^{2}} = \frac{4}{9} \]

### Final Answer
Thus, the final result of the expression \((\frac{-2}{3})^{10} \div (\frac{-2}{3})^{8}\) is:
\[ \frac{4}{9} \]

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