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: \(-x^{2}\ z^{5}\)
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: \(-x^{2}\ z^{5}\)
Answers
GPT-4o mini
Let's go through each of the problems, explaining why the answers are correct and solving them step by step.
### Problem 1
**Statement**: Using the Product Rule of Exponents, what is an equivalent form of \(9^{5} \cdot 9^{2}\)?
**Rule Used**: The Product Rule of Exponents states that when we multiply two powers with the same base, we add the exponents. The formula can be expressed as:
\[
a^{m} \cdot a^{n} = a^{m+n}
\]
**Solution**:
1. Identify the base and the exponents:
- Base: \(9\)
- Exponents: \(5\) and \(2\)
2. Apply the Product Rule:
- Using the formula: \(9^{5} \cdot 9^{2} = 9^{5+2}\)
3. Calculate the sum of the exponents:
- \(5 + 2 = 7\)
4. Thus, we can rewrite the expression:
- \(9^{5} \cdot 9^{2} = 9^{7}\)
**Final Answer**: Hence, the equivalent form of \(9^{5} \cdot 9^{2}\) is **\(9^{7}\)**.
---
### Problem 2
**Statement**: Apply the Power of Zero Rule of exponents to simplify \(-4^{0}x^{2}y^{0}z^{5}\).
**Rule Used**: The Power of Zero Rule states that any non-zero number raised to the power of zero is equal to 1. Formally, \(a^{0} = 1\) for any \(a \neq 0\).
**Solution**:
1. Evaluate each part of the expression:
- \(-4^{0}\) = \(-1\) because \(4^{0} = 1\)
- \(x^{2}\) remains as it is.
- \(y^{0} = 1\) (since zero exponent rule applies).
- \(z^{5}\) remains as it is.
2. Substitute the evaluated parts into the expression:
- The expression \(-4^{0} \cdot x^{2} \cdot y^{0} \cdot z^{5}\) becomes:
- \(-1 \cdot x^{2} \cdot 1 \cdot z^{5}\)
3. Simplify the expression:
- Ignoring the multiplied 1 gives us:
- \(-1 \cdot x^{2} \cdot z^{5} = -x^{2}z^{5}\)
**Final Answer**: Therefore, after applying the Power of Zero Rule, the simplified expression is **\(-x^{2} z^{5}\)**.
### Problem 1
**Statement**: Using the Product Rule of Exponents, what is an equivalent form of \(9^{5} \cdot 9^{2}\)?
**Rule Used**: The Product Rule of Exponents states that when we multiply two powers with the same base, we add the exponents. The formula can be expressed as:
\[
a^{m} \cdot a^{n} = a^{m+n}
\]
**Solution**:
1. Identify the base and the exponents:
- Base: \(9\)
- Exponents: \(5\) and \(2\)
2. Apply the Product Rule:
- Using the formula: \(9^{5} \cdot 9^{2} = 9^{5+2}\)
3. Calculate the sum of the exponents:
- \(5 + 2 = 7\)
4. Thus, we can rewrite the expression:
- \(9^{5} \cdot 9^{2} = 9^{7}\)
**Final Answer**: Hence, the equivalent form of \(9^{5} \cdot 9^{2}\) is **\(9^{7}\)**.
---
### Problem 2
**Statement**: Apply the Power of Zero Rule of exponents to simplify \(-4^{0}x^{2}y^{0}z^{5}\).
**Rule Used**: The Power of Zero Rule states that any non-zero number raised to the power of zero is equal to 1. Formally, \(a^{0} = 1\) for any \(a \neq 0\).
**Solution**:
1. Evaluate each part of the expression:
- \(-4^{0}\) = \(-1\) because \(4^{0} = 1\)
- \(x^{2}\) remains as it is.
- \(y^{0} = 1\) (since zero exponent rule applies).
- \(z^{5}\) remains as it is.
2. Substitute the evaluated parts into the expression:
- The expression \(-4^{0} \cdot x^{2} \cdot y^{0} \cdot z^{5}\) becomes:
- \(-1 \cdot x^{2} \cdot 1 \cdot z^{5}\)
3. Simplify the expression:
- Ignoring the multiplied 1 gives us:
- \(-1 \cdot x^{2} \cdot z^{5} = -x^{2}z^{5}\)
**Final Answer**: Therefore, after applying the Power of Zero Rule, the simplified expression is **\(-x^{2} z^{5}\)**.