Let's focus on solving the problems you’ve shared in the image, starting with one at a time.

Problem 5:

$16^{4.5} \times 4^{6.3} \times 8^{2.1} \div 2^{9.2} \times 32^{0.64} = 8^{2.3 + ?}$

We can rewrite everything in terms of base $2$ to simplify this equation.

• $16 = 2^4$
• $4 = 2^2$
• $8 = 2^3$
• $32 = 2^5$

Now, express all terms in base 2:

$16^{4.5} = (2^4)^{4.5} = 2^{18}, \quad 4^{6.3} = (2^2)^{6.3} = 2^{12.6}, \quad 8^{2.1} = (2^3)^{2.1} = 2^{6.3}$ $2^{9.2} = 2^{9.2}, \quad 32^{0.64} = (2^5)^{0.64} = 2^{3.2}$

Now combine the terms: $2^{18} \times 2^{12.6} \times 2^{6.3} \div 2^{9.2} \times 2^{3.2}$ $= 2^{(18 + 12.6 + 6.3 - 9.2 + 3.2)} = 2^{30.9}$

Now, set it equal to the right-hand side: $2^{30.9} = 8^{2.3 + ?} = (2^3)^{2.3 + ?} = 2^{3(2.3 + ?)}$

Equating the exponents: $30.9 = 3(2.3 + ?)$ $2.3 + ? = \frac{30.9}{3} = 10.3$ $? = 10.3 - 2.3 = 8$

Thus, the solution to Problem 5 is $? = 8$.

Would you like to go over the next problem (Problem 6)?

Here are some related questions to explore this topic further:

1. How do you simplify expressions involving exponents and powers?
2. What are the properties of exponents when multiplying or dividing terms with the same base?
3. How do you express numbers like 16, 8, and 32 as powers of 2 for simplification?
4. What is the general method for solving equations involving exponents?
5. How can logarithms help solve equations involving exponents?

Tip: When working with powers, always try to express everything in terms of a common base to simplify the calculations!