The mathematical expression in the image is:

$16^{\frac{3}{4}n} \div 8^{\frac{5}{3}n} \times 4^{n+1}$

Simplification Steps

Let's simplify this expression step by step.

1. Express each base in terms of powers of 2:

   • $16 = 2^4$, so $16^{\frac{3}{4}n} = (2^4)^{\frac{3}{4}n} = 2^{3n}$.
   • $8 = 2^3$, so $8^{\frac{5}{3}n} = (2^3)^{\frac{5}{3}n} = 2^{5n}$.
   • $4 = 2^2$, so $4^{n+1} = (2^2)^{n+1} = 2^{2(n+1)} = 2^{2n + 2}$.

2. Rewrite the expression using these powers of 2:

$2^{3n} \div 2^{5n} \times 2^{2n+2}$

3. Apply the laws of exponents:

   • Division: $\frac{2^a}{2^b} = 2^{a-b}$
   • Multiplication: $2^a \times 2^b = 2^{a+b}$

So, we have:

$2^{3n - 5n} \times 2^{2n+2} = 2^{-2n} \times 2^{2n+2}$

4. Combine the exponents:

$2^{-2n + 2n + 2} = 2^{2}$

5. Final simplified expression:

$2^2 = 4$

Conclusion

The simplified result of the expression is 4.

Would you like more details, or do you have any questions about this solution?

Related Questions:

1. How do we simplify expressions involving exponents with different bases?
2. Can we simplify the given expression using logarithms?
3. What are some other examples where we use the laws of exponents in algebra?
4. How can we apply these rules to solve exponential equations?
5. How do exponent rules change when we deal with negative or fractional exponents?

Tip:

Always try to express numbers in terms of their prime bases when simplifying expressions with exponents. This makes it easier to apply exponent rules effectively.