To write each expression as a single power and then evaluate, we can use the following property of exponents:

$a^m \times a^n = a^{m+n}$

Let's apply this to each expression:

a) $5^2 \times 5^2$

$5^2 \times 5^2 = 5^{2+2} = 5^4$

Evaluate:

$5^4 = 625$

b) $2^4 \times 2^3$

$2^4 \times 2^3 = 2^{4+3} = 2^7$

Evaluate:

$2^7 = 128$

c) $(-3)^2 \times (-3)^4$

$(-3)^2 \times (-3)^4 = (-3)^{2+4} = (-3)^6$

Evaluate:

$(-3)^6 = 729$

d) $(-4)^3 \times (-4)^3$

$(-4)^3 \times (-4)^3 = (-4)^{3+3} = (-4)^6$

Evaluate:

$(-4)^6 = 4096$

e) $\left(\frac{1}{4}\right)^2 \times \left(\frac{1}{4}\right)^3$

$\left(\frac{1}{4}\right)^2 \times \left(\frac{1}{4}\right)^3 = \left(\frac{1}{4}\right)^{2+3} = \left(\frac{1}{4}\right)^5$

Evaluate:

$\left(\frac{1}{4}\right)^5 = \frac{1}{1024}$

f) $\left(-\frac{1}{2}\right)^2 \times \left(-\frac{1}{2}\right)^1$

$\left(-\frac{1}{2}\right)^2 \times \left(-\frac{1}{2}\right)^1 = \left(-\frac{1}{2}\right)^{2+1} = \left(-\frac{1}{2}\right)^3$

Evaluate:

$\left(-\frac{1}{2}\right)^3 = -\frac{1}{8}$

Would you like more details on any of these steps, or do you have any further questions?

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Here are 5 related questions you might find helpful:

1. How do you multiply powers with the same base when the exponents are different?
2. What happens when you multiply two negative numbers with even exponents?
3. Can you explain why $(-3)^6$ results in a positive number?
4. How do you handle fractional exponents when multiplying powers?
5. What are the general rules for working with powers of fractions?

Tip: Always check if the bases are the same before applying exponent rules when multiplying or dividing powers.