Question

Explain Why these answers are correct and solve them.
1. Using the Product Rule Of Exponents, what is an equivelant form of \(9^{5}\cdot 9^{2}\). Answer: \(9^{7}\

2. Apply the Power of Zero Rule of exponents to simplify \(-4^{0}x^{2}y^{0}z^{5}\).
Answer: \(-x^{2}\ z^{5}\)

Answer 1

Let's go through each of your queries step by step.

### 1. Product Rule of Exponents

**Statement**: We want to simplify the expression \(9^{5} \cdot 9^{2}\).

**Explanation**:
The Product Rule of Exponents states that when you multiply two powers with the same base, you can add the exponents. Algebraically, this is expressed as:
\[
a^{m} \cdot a^{n} = a^{m+n}
\]
Where \(a\) is the base and \(m\) and \(n\) are the exponents.

**Application**:
In your case:
\[
9^{5} \cdot 9^{2} = 9^{5+2} = 9^{7}
\]

Thus, the equivalent form of \(9^{5} \cdot 9^{2}\) is indeed \(9^{7}\).

### 2. Power of Zero Rule and Simplification

**Statement**: We need to simplify the expression \(-4^{0}x^{2}y^{0}z^{5}\).

**Explanation**:
The Power of Zero Rule states that any non-zero number raised to the power of zero equals 1. This can be written as:
\[
a^{0} = 1 \quad (\text{for } a \neq 0)
\]

In this expression, it is important to note:
- \(4^{0} = 1\) (because 4 is non-zero),
- \(y^{0} = 1\) (because \(y\) is also a variable that can take non-zero values).

**Application**:
Using these rules, we can simplify:
\[
-4^{0}x^{2}y^{0}z^{5} = -1 \cdot x^{2} \cdot 1 \cdot z^{5} = -x^{2}z^{5}
\]

Therefore, the simplified form of \(-4^{0}x^{2}y^{0}z^{5}\) is indeed \(-x^{2}z^{5}\).

### Summary of Answers

1. \(9^{5} \cdot 9^{2} = 9^{7}\) (using the Product Rule of Exponents).
2. \(-4^{0}x^{2}y^{0}z^{5} = -x^{2}z^{5}\) (applying the Power of Zero Rule).