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### 1. Simplify the following expressions:
#### i. \( x^8 \div x^8 \)
Using the division of powers property, \( a^m \div a^n = a^{m-n} \):
\[
x^8 \div x^8 = x^{8-8} = x^0 = 1
\]

#### ii. \( (2p)^4 \times (2p)^{-4} \)
Use the property of exponents \( (a^m)(a^n) = a^{m+n} \):
\[
(2p)^4 \times (2p)^{-4} = (2p)^{4 + (-4)} = (2p)^0 = 1
\]

#### iii. \( \frac{a^2 \times a^3}{a \times a^4} \)
Simplify using the exponent rules:
\[
\frac{a^2 \times a^3}{a \times a^4} = \frac{a^{2+3}}{a^{1+4}} = \frac{a^5}{a^5} = a^0 = 1
\]

#### iv. \( \frac{y^4 \times y^2}{y^6} \)
Using the division of powers:
\[
\frac{y^4 \times y^2}{y^6} = \frac{y^{4+2}}{y^6} = \frac{y^6}{y^6} = y^0 = 1
\]

#### v. \( \frac{p^3 \times p^5 \times p}{p^6 \times p^3} \)
Using the multiplication and division of powers:
\[
\frac{p^3 \times p^5 \times p}{p^6 \times p^3} = \frac{p^{3+5+1}}{p^{6+3}} = \frac{p^9}{p^9} = p^0 = 1
\]

#### vi. \( \frac{x^{-2} \times x^{-4} \times x^6}{y^{-2} \times y^3 \times y^{-6}} \)
Simplify each part:
\[
\frac{x^{-2} \times x^{-4} \times x^6}{y^{-2} \times y^3 \times y^{-6}} = \frac{x^{-2-4+6}}{y^{-2+3-6}} = \frac{x^0}{y^{-5}} = \frac{1}{y^{-5}} = y^5
\]

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### 2. Evaluate the following:
#### i. \( 2^0 \times 3 \)
Since \( 2^0 = 1 \):
\[
2^0 \times 3 = 1 \times 3 = 3
\]

#### ii. \( (-4)^0 \)
Any non-zero number raised to the power of 0 equals 1:
\[
(-4)^0 = 1
\]

#### iii. \( \left( \frac{x}{y} \right)^0 + 1 \)
Any expression raised to the power of 0 equals 1:
\[
\left( \frac{x}{y} \right)^0 + 1 = 1 + 1 = 2
\]

#### iv. \( \left( \frac{x^2}{y^2} \right)^0 \)
\[
\left( \frac{x^2}{y^2} \right)^0 = 1
\]

#### v. \( 5^0 + 1 \)
Since \( 5^0 = 1 \):
\[
5^0 + 1 = 1 + 1 = 2
\]

#### vi. \( \left( \frac{2}{3} \right)^0 \)
\[
\left( \frac{2}{3} \right)^0 = 1
\]

#### vii. \( (2ab)^0 - 2^0 \)
\[
(2ab)^0 - 2^0 = 1 - 1 = 0
\]

#### viii. \( (abc)^0 \)
\[
(abc)^0 = 1
\]

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Would you like further clarification or details on any of these solutions?

### 5 Related Questions:
1. What happens when a negative base is raised to an even exponent?
2. How do you handle division with variables that have negative exponents?
3. What is the result of multiplying terms with the same base but different exponents?
4. What is the importance of the zero exponent rule in algebra?
5. How can we simplify expressions involving both powers and roots?

### Tip:
Always remember that any non-zero number or expression raised to the power of 0 is equal to 1, which can simplify many complex expressions.

Simplify and evaluate the given mathematical expressions.

Learn how to simplify and evaluate exponential expressions using rules for exponents. This lesson covers key concepts such as division of powers, the zero exponent rule, and multiplication of powers. Suitable for students in Grades 8-10.

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