Let's rewrite each expression as a single power of 2.

a. $4^{10}$:

• $4$ is $2^2$, so: $4^{10} = (2^2)^{10} = 2^{20}$

b. $8^x \times 2^4$:

• $8$ is $2^3$, so: $8^x \times 2^4 = (2^3)^x \times 2^4 = 2^{3x} \times 2^4 = 2^{3x+4}$

c. $16^{\frac{2}{3}}$:

• $16$ is $2^4$, so: $16^{\frac{2}{3}} = (2^4)^{\frac{2}{3}} = 2^{\frac{8}{3}}$

d. $2^x \times 4^y \times 8^z \times 16$:

• Rewrite $4$, $8$, and $16$ in terms of powers of 2: $2^x \times (2^2)^y \times (2^3)^z \times 2^4 = 2^x \times 2^{2y} \times 2^{3z} \times 2^4 = 2^{x+2y+3z+4}$

e. $2\sqrt{2}$:

• $\sqrt{2} = 2^{1/2}$, so: $2\sqrt{2} = 2^1 \times 2^{1/2} = 2^{1 + \frac{1}{2}} = 2^{\frac{3}{2}}$

f. $\frac{4}{\sqrt{2}}$:

• Rewrite $4$ and $\sqrt{2}$ as powers of 2: $\frac{4}{\sqrt{2}} = \frac{2^2}{2^{1/2}} = 2^{2 - \frac{1}{2}} = 2^{\frac{3}{2}}$

g. $4^{1/3} \times 8^{1/5}$:

• Rewrite $4$ and $8$ as powers of 2: $4^{1/3} \times 8^{1/5} = (2^2)^{1/3} \times (2^3)^{1/5} = 2^{\frac{2}{3}} \times 2^{\frac{3}{5}} = 2^{\frac{2}{3} + \frac{3}{5}} = 2^{\frac{19}{15}}$

Summary:

a. $2^{20}$
b. $2^{3x + 4}$
c. $2^{\frac{8}{3}}$
d. $2^{x + 2y + 3z + 4}$
e. $2^{\frac{3}{2}}$
f. $2^{\frac{3}{2}}$
g. $2^{\frac{19}{15}}$

Would you like more details on any step?

Here are five related questions:

1. How do you simplify expressions with fractional exponents?
2. What are the general properties of exponents?
3. How do you handle exponents when multiplying or dividing terms?
4. How would you express roots as fractional powers?
5. How do you combine terms with different bases and exponents?

Tip: Always express roots and radicals in terms of fractional exponents for easier simplification.