The problem asks to simplify the exponents in two expressions:

a. $5^2 \times 5^3$

Since the base is the same (5), you can apply the rule of exponents for multiplication:

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

Thus,

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

b. $\left(\frac{1}{3}\right)^5 \times \left(\frac{1}{3}\right)^2$

Similarly, because the base is the same $\frac{1}{3}$, apply the same rule of exponents:

$\left(\frac{1}{3}\right)^5 \times \left(\frac{1}{3}\right)^2 = \left(\frac{1}{3}\right)^{5+2} = \left(\frac{1}{3}\right)^7$

So the simplified expressions are:

• a. $5^5$
• b. $\left(\frac{1}{3}\right)^7$

Would you like further explanation or details on exponent rules?

Here are 5 related questions you could explore:

1. What are the properties of exponents for division?
2. How do negative exponents affect the value of an expression?
3. What is the rule for raising a power to another power?
4. How do fractional exponents relate to radicals (square roots, cube roots, etc.)?
5. What happens if the base of the exponents is negative?

Tip: When multiplying or dividing exponents with the same base, you can simplify by adding or subtracting the exponents, respectively.