If `x= (2/3)^(4)div (2/3)^(2)`, then find the value of `x^(2)+2x+3`.

Answer

To solve the problem step by step, we start with the given expression:

**Step 1: Simplify \( x \)**
We have:
\[ x = \frac{(2/3)^4}{(2/3)^2} \]
Using the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \), we can rewrite this as:
\[ x = (2/3)^{4-2} = (2/3)^2 \]

**Step 2: Calculate \( x \)**
Now, we can calculate \( (2/3)^2 \):
\[ x = \frac{2^2}{3^2} = \frac{4}{9} \]

**Step 3: Substitute \( x \) into the expression \( x^2 + 2x + 3 \)**
Next, we need to find the value of \( x^2 + 2x + 3 \):
\[ x^2 + 2x + 3 = \left(\frac{4}{9}\right)^2 + 2\left(\frac{4}{9}\right) + 3 \]

**Step 4: Calculate \( x^2 \)**
Calculating \( \left(\frac{4}{9}\right)^2 \):
\[ \left(\frac{4}{9}\right)^2 = \frac{16}{81} \]

**Step 5: Calculate \( 2x \)**
Now calculate \( 2 \times \frac{4}{9} \):
\[ 2 \times \frac{4}{9} = \frac{8}{9} \]

**Step 6: Combine the terms**
Now we can combine all the terms:
\[ x^2 + 2x + 3 = \frac{16}{81} + \frac{8}{9} + 3 \]
To add these fractions, we need a common denominator. The common denominator for 81 and 9 is 81.
Convert \( \frac{8}{9} \) and \( 3 \) to have a denominator of 81:
\[ \frac{8}{9} = \frac{8 \times 9}{9 \times 9} = \frac{72}{81} \]
\[ 3 = \frac{3 \times 81}{1 \times 81} = \frac{243}{81} \]

**Step 7: Add the fractions**
Now we can add:
\[ x^2 + 2x + 3 = \frac{16}{81} + \frac{72}{81} + \frac{243}{81} = \frac{16 + 72 + 243}{81} = \frac{331}{81} \]

Thus, the final answer is:
\[ x^2 + 2x + 3 = \frac{331}{81} \]

---