To evaluate each expression in two different ways, we can either simplify using the properties of exponents or directly compute the values. Let's go through each problem.

a) $6^2 \times 6^3$

Method 1: Use the product rule for exponents:

$6^2 \times 6^3 = 6^{2+3} = 6^5$ $6^5 = 7776$

Method 2: Calculate each power separately and multiply:

$6^2 = 36 \quad \text{and} \quad 6^3 = 216$ $36 \times 216 = 7776$

b) $7^4 \times 7^2$

Method 1: Use the product rule for exponents:

$7^4 \times 7^2 = 7^{4+2} = 7^6$ $7^6 = 117649$

Method 2: Calculate each power separately and multiply:

$7^4 = 2401 \quad \text{and} \quad 7^2 = 49$ $2401 \times 49 = 117649$

c) $\frac{9^5}{9^3}$

Method 1: Use the quotient rule for exponents:

$\frac{9^5}{9^3} = 9^{5-3} = 9^2$ $9^2 = 81$

Method 2: Calculate each power separately and divide:

$9^5 = 59049 \quad \text{and} \quad 9^3 = 729$ $\frac{59049}{729} = 81$

d) $\frac{(-7)^4}{(-7)^3}$

Method 1: Use the quotient rule for exponents:

$\frac{(-7)^4}{(-7)^3} = (-7)^{4-3} = (-7)^1 = -7$

Method 2: Calculate each power separately and divide:

$(-7)^4 = 2401 \quad \text{and} \quad (-7)^3 = -343$ $\frac{2401}{-343} = -7$

e) $(5^2)^3$

Method 1: Use the power rule for exponents:

$(5^2)^3 = 5^{2 \times 3} = 5^6$ $5^6 = 15625$

Method 2: Calculate the inner power first, then raise to the outer power:

$5^2 = 25$ $25^3 = 15625$

f) $(10^5)^2$

Method 1: Use the power rule for exponents:

$(10^5)^2 = 10^{5 \times 2} = 10^{10}$ $10^{10} = 10000000000$

Method 2: Calculate the inner power first, then raise to the outer power:

$10^5 = 100000$ $100000^2 = 10000000000$

g) $(-8)^3 \times (-8)$

Method 1: Combine the exponents:

$(-8)^3 \times (-8) = (-8)^3 \times (-8)^1 = (-8)^{3+1} = (-8)^4$ $(-8)^4 = 4096$

Method 2: Calculate each power separately and multiply:

$(-8)^3 = -512 \quad \text{and} \quad (-8)^1 = -8$ $-512 \times -8 = 4096$

h) $[(-1)^{11}]^9$

Method 1: Simplify the inner expression first:

$(-1)^{11} = -1 \quad \text{so} \quad (-1^{11})^9 = (-1)^9$ $(-1)^9 = -1$

Method 2: Use the power rule for exponents:

$[(-1)^{11}]^9 = (-1)^{11 \times 9} = (-1)^{99}$ $(-1)^{99} = -1$

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Do you want more details on any step or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the product rule of exponents, and why does it apply in these cases?
2. How do we handle negative bases with exponents?
3. What is the quotient rule for exponents, and how can it simplify expressions?
4. How do you compute powers of large numbers without a calculator?
5. What happens when the exponent of a negative number is even vs. odd?

Tip: Always check the sign when dealing with negative bases and odd or even exponents!