show how 48 * 4^(2n-1) + 6 * 2^(4n+1) to 3 * 2(4n+3)

To simplify the expression \(48 \times 4^{(2n-1)} + 6 \times 2^{(4n+1)}\) to the form \(3 \times 2^{(4n+3)}\), let's follow the steps carefully.

### Step 1: Rewrite the powers of 4 in terms of powers of 2.
We know that \(4 = 2^2\), so we can express powers of 4 in terms of powers of 2:

\[
4^{(2n-1)} = (2^2)^{(2n-1)} = 2^{2(2n-1)} = 2^{4n-2}
\]

Thus, the first term becomes:

\[
48 \times 4^{(2n-1)} = 48 \times 2^{4n-2}
\]

### Step 2: Write down the expression with both terms using powers of 2.
The original expression becomes:

\[
48 \times 2^{4n-2} + 6 \times 2^{4n+1}
\]

### Step 3: Factor out the common factor.
The two terms \(48 \times 2^{4n-2}\) and \(6 \times 2^{4n+1}\) have a common factor of \(2^{4n-2}\). Let's factor that out:

\[
48 \times 2^{4n-2} + 6 \times 2^{4n+1} = 2^{4n-2} \times (48 + 6 \times 2^3)
\]

### Step 4: Simplify the expression inside the parentheses.
We know that \(2^3 = 8\), so:

\[
48 + 6 \times 8 = 48 + 48 = 96
\]

Now, the expression becomes:

\[
2^{4n-2} \times 96
\]

### Step 5: Break down the constant factor.
We can express 96 as \(96 = 3 \times 2^5\). So, substitute this into the equation:

\[
2^{4n-2} \times 96 = 2^{4n-2} \times 3 \times 2^5
\]

### Step 6: Combine the powers of 2.
We can combine the powers of 2:

\[
2^{4n-2} \times 2^5 = 2^{(4n-2+5)} = 2^{4n+3}
\]

Thus, the expression simplifies to:

\[
3 \times 2^{4n+3}
\]

### Final Answer:
So, \(48 \times 4^{2n-1} + 6 \times 2^{4n+1}\) simplifies to:

\[
3 \times 2^{4n+3}
\]

Would you like further clarification or details on any step?

### Relative Questions:
1. What is the general rule for converting powers of 4 to powers of 2?
2. How does factoring out the greatest common factor help in simplification?
3. How do you combine exponents when multiplying terms with the same base?
4. Can you express \(4^{3n+2}\) in terms of powers of 2?
5. How would you approach factoring when dealing with polynomials?

### Tip:
Always look for common factors or patterns (like powers of 2) when simplifying algebraic expressions to make the process easier.

Learn how to simplify the expression 48 * 4^(2n-1) + 6 * 2^(4n+1) using properties of exponents and factorization. This detailed step-by-step solution converts powers of 4 to powers of 2 and factors out common factors for simplification. Suitable for high school students studying algebra and exponentiation.

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