is 2^x + (((2^x)-1)/3)(((2^y)-1)/3) equal to (((2^(x+y))-1)/3)?

Let's analyze the given expression step by step to determine if \(2^x + \frac{(2^x - 1)}{3} \cdot \frac{(2^y - 1)}{3}\) is equal to \(\frac{2^{x+y} - 1}{3}\).

1. **Left-hand side (LHS)**: 

\[ 2^x + \frac{(2^x - 1)}{3} \cdot \frac{(2^y - 1)}{3} \]

First, simplify the product inside the LHS:
\[ \frac{(2^x - 1)}{3} \cdot \frac{(2^y - 1)}{3} = \frac{(2^x - 1)(2^y - 1)}{9} \]

So, the LHS becomes:
\[ 2^x + \frac{(2^x - 1)(2^y - 1)}{9} \]

2. **Right-hand side (RHS)**:
\[ \frac{2^{x+y} - 1}{3} \]

We need to see if the LHS can be manipulated to look like the RHS.

3. **Expanding the LHS**:
\[ 2^x + \frac{(2^x - 1)(2^y - 1)}{9} \]

Expanding the product \((2^x - 1)(2^y - 1)\):
\[ (2^x - 1)(2^y - 1) = 2^x \cdot 2^y - 2^x - 2^y + 1 = 2^{x+y} - 2^x - 2^y + 1 \]

Substitute this back into the LHS expression:
\[ 2^x + \frac{2^{x+y} - 2^x - 2^y + 1}{9} \]

4. **Combining terms**:
To match the form of the RHS, multiply the entire fraction by 3 to make the denominators equal:
\[ \frac{9 \cdot 2^x}{9} + \frac{2^{x+y} - 2^x - 2^y + 1}{9} = \frac{9 \cdot 2^x + 2^{x+y} - 2^x - 2^y + 1}{9} \]

Combine the terms in the numerator:
\[ \frac{8 \cdot 2^x + 2^{x+y} - 2^y + 1}{9} \]

5. **Comparing with RHS**:
For the LHS to equal the RHS, the numerator must match the numerator of the RHS multiplied by 3:
\[ 3 \cdot (2^{x+y} - 1) = 3 \cdot 2^{x+y} - 3 \]

So, the numerator on the RHS is:
\[ 2^{x+y} - 1 \]

This does not match the numerator we derived for the LHS:
\[ 8 \cdot 2^x + 2^{x+y} - 2^y + 1 \]

Thus, the two expressions are not equal because the terms on the left-hand side do not simplify to the form required by the right-hand side.

Therefore, 

\[ 2^x + \frac{(2^x - 1)}{3} \cdot \frac{(2^y - 1)}{3} \neq \frac{2^{x+y} - 1}{3} \]

Would you like any more details on this solution or have any questions? Here are some related questions you might be interested in:

1. How to simplify complex fractions involving exponential terms?
2. What are the common properties of exponents?
3. How to factor expressions involving exponents?
4. What are the steps for solving exponential equations?
5. How to verify if two algebraic expressions are equivalent?

**Tip:** When dealing with exponentiation and fractional expressions, always try to simplify step by step and check if you can factor out common terms.

Explore the equivalence problem involving exponents and fractional expressions: is 2^x + (((2^x) - 1)/3)(((2^y) - 1)/3) equal to (((2^(x+y)) - 1)/3)? Learn the step-by-step solution and detailed analysis to understand why these expressions are not equal. Suitable for advanced high school students and anyone interested in algebraic manipulation of exponential terms.

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