Answer :
Let's analyze the assertion and the reason step-by-step.
Assertion: [tex]\(3^5 \times 3^7 = 3^{12}\)[/tex]
To verify the assertion, we use the properties of exponents. Specifically, the property [tex]\(a^m \times a^n = a^{m+n}\)[/tex] where [tex]\(a\)[/tex] is the base and [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are the exponents.
Applying this to the assertion:
[tex]\[ 3^5 \times 3^7 = 3^{5+7} = 3^{12} \][/tex]
So the assertion is true.
Reason: [tex]\((a^m)^n = a^{m \times n}\)[/tex]
To verify the reason, let's use the property directly. It states that when you have a power of a power, you multiply the exponents.
For example:
[tex]\[ (a^m)^n = a^{m \times n} \][/tex]
This is a fundamental property of exponents and it is a true mathematical statement. Therefore, the reason is true.
Next, determine if the reason is the correct explanation of the assertion.
The assertion [tex]\(3^5 \times 3^7 = 3^{12}\)[/tex] is verified using the property [tex]\(a^m \times a^n = a^{m+n}\)[/tex] which is a different property from the one stated in the reason [tex]\((a^m)^n = a^{m \times n}\)[/tex]. Although both the assertion and the reason are true, the reason does not explain why the assertion is true because it involves a different property of exponents.
Conclusion:
- Both the assertion and the reason are true.
- The reason is not the correct explanation of the assertion.
Thus, the correct choice is:
b) Both assertion and reason are true and reason is not the correct explanation of assertion.
Assertion: [tex]\(3^5 \times 3^7 = 3^{12}\)[/tex]
To verify the assertion, we use the properties of exponents. Specifically, the property [tex]\(a^m \times a^n = a^{m+n}\)[/tex] where [tex]\(a\)[/tex] is the base and [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are the exponents.
Applying this to the assertion:
[tex]\[ 3^5 \times 3^7 = 3^{5+7} = 3^{12} \][/tex]
So the assertion is true.
Reason: [tex]\((a^m)^n = a^{m \times n}\)[/tex]
To verify the reason, let's use the property directly. It states that when you have a power of a power, you multiply the exponents.
For example:
[tex]\[ (a^m)^n = a^{m \times n} \][/tex]
This is a fundamental property of exponents and it is a true mathematical statement. Therefore, the reason is true.
Next, determine if the reason is the correct explanation of the assertion.
The assertion [tex]\(3^5 \times 3^7 = 3^{12}\)[/tex] is verified using the property [tex]\(a^m \times a^n = a^{m+n}\)[/tex] which is a different property from the one stated in the reason [tex]\((a^m)^n = a^{m \times n}\)[/tex]. Although both the assertion and the reason are true, the reason does not explain why the assertion is true because it involves a different property of exponents.
Conclusion:
- Both the assertion and the reason are true.
- The reason is not the correct explanation of the assertion.
Thus, the correct choice is:
b) Both assertion and reason are true and reason is not the correct explanation of assertion.
Answer:Let's evaluate the assertion and reason provided:
**Assertion:** \( 3^5 \cdot 3^7 = 3^{12} \)
**Reason:** \( (a^m)^n = a^{m \cdot n} \)
Let's analyze each part:
### Assertion Analysis:
The assertion states \( 3^5 \cdot 3^7 = 3^{12} \).
This can be verified using the properties of exponents:
\[ 3^5 \cdot 3^7 = 3^{5+7} = 3^{12} \]
So, the assertion \( 3^5 \cdot 3^7 = 3^{12} \) is **true**.
### Reason Analysis:
The reason given is \( (a^m)^n = a^{m \cdot n} \).
This is the correct exponentiation property:
\[ (a^m)^n = a^{m \cdot n} \]
Therefore, the reason \( (a^m)^n = a^{m \cdot n} \) is **true**.
### Evaluation of Options:
- Option a) Both assertion and reason are true, and the reason is the correct explanation of the assertion.
- The assertion and reason are both true.
- The reason correctly explains why the assertion is true (using the exponentiation property).
Hence, the correct answer is **a)** Both assertion and reason are true, and the reason is the correct explanation of the assertion.
Step-by-step explanation:Certainly! Let's break down the assertion and reason step-by-step:
**Assertion:** \( 3^5 \cdot 3^7 = 3^{12} \)
To verify the assertion:
1. **Exponentiation Rule:**
According to the exponentiation rule \( a^m \cdot a^n = a^{m+n} \),
\[ 3^5 \cdot 3^7 = 3^{5+7} \]
2. **Calculate the Exponent:**
\[ 5 + 7 = 12 \]
3. **Conclusion:**
Therefore, \( 3^5 \cdot 3^7 = 3^{12} \) is true. This means the assertion itself is true.
**Reason:** \( (a^m)^n = a^{m \cdot n} \)
To evaluate the reason:
1. **Exponentiation Rule:**
The reason states the exponentiation property \( (a^m)^n = a^{m \cdot n} \).
2. **Verify the Property:**
Let's apply this property to \( 3^5 \):
\[ (3^5)^7 = 3^{5 \cdot 7} \]
3. **Calculate the Exponent:**
\[ 5 \cdot 7 = 35 \]
4. **Conclusion:**
Therefore, \( (3^5)^7 = 3^{35} \), which matches \( 3^{5 \cdot 7} = 3^{12} \). This confirms that the reason \( (a^m)^n = a^{m \cdot n} \) is true.
### Analysis of Options:
- **Option a) Both assertion and reason are true, and the reason is the correct explanation of the assertion:**
- The assertion \( 3^5 \cdot 3^7 = 3^{12} \) is true.
- The reason \( (a^m)^n = a^{m \cdot n} \) is true and correctly explains why the assertion is true (by applying the exponentiation property).
Since both the assertion and reason are individually true, and the reason correctly explains why the assertion holds based on the exponentiation property, **option a)** is the correct answer.
This completes the step-by-step explanation verifying the correctness of the assertion and reason provided.
