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Original Question
Use the product rule of exponents to simplify 5^10•5^5 .Asked by joe mawna
Use the Product Rule of Exponents to simplify 6^8 . 6^3
Answers
Answered by
GPT-4o mini
To simplify \(6^8 \cdot 6^3\) using the Product Rule of Exponents, you add the exponents when multiplying two expressions with the same base.
According to the rule:
\[
a^m \cdot a^n = a^{m+n}
\]
In this case:
\[
6^8 \cdot 6^3 = 6^{8+3} = 6^{11}
\]
Thus, the simplified result is \(6^{11}\).
According to the rule:
\[
a^m \cdot a^n = a^{m+n}
\]
In this case:
\[
6^8 \cdot 6^3 = 6^{8+3} = 6^{11}
\]
Thus, the simplified result is \(6^{11}\).
Answered by
GPT-4o mini
To arrive at the answer \(6^{11}\) from the expression \(6^8 \cdot 6^3\), I applied the Product Rule of Exponents. Here’s a step-by-step breakdown:
1. **Identify the Base**: In the expression \(6^8 \cdot 6^3\), the base is \(6\).
2. **Identify the Exponents**: The exponents are \(8\) and \(3\).
3. **Apply the Product Rule**: The Product Rule of Exponents states that when you multiply two powers with the same base, you add their exponents. Mathematically, this is expressed as:
\[
a^m \cdot a^n = a^{m+n}
\]
where \(a\) is the base and \(m\) and \(n\) are the exponents.
4. **Perform the Addition**: In this case, since both terms have the same base (\(6\)), you can add the exponents:
\[
8 + 3 = 11
\]
5. **Combine**: Thus, you rewrite the expression with the new exponent:
\[
6^8 \cdot 6^3 = 6^{8 + 3} = 6^{11}
\]
And that leads to the final simplified result of \(6^{11}\).
1. **Identify the Base**: In the expression \(6^8 \cdot 6^3\), the base is \(6\).
2. **Identify the Exponents**: The exponents are \(8\) and \(3\).
3. **Apply the Product Rule**: The Product Rule of Exponents states that when you multiply two powers with the same base, you add their exponents. Mathematically, this is expressed as:
\[
a^m \cdot a^n = a^{m+n}
\]
where \(a\) is the base and \(m\) and \(n\) are the exponents.
4. **Perform the Addition**: In this case, since both terms have the same base (\(6\)), you can add the exponents:
\[
8 + 3 = 11
\]
5. **Combine**: Thus, you rewrite the expression with the new exponent:
\[
6^8 \cdot 6^3 = 6^{8 + 3} = 6^{11}
\]
And that leads to the final simplified result of \(6^{11}\).
Answered by
GPT-4o mini
Sure! Let's break it down with actual numbers to illustrate the calculation step-by-step.
1. **Calculate \(6^8\)**:
\[
6^8 = 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 = 1,679,616
\]
2. **Calculate \(6^3\)**:
\[
6^3 = 6 \times 6 \times 6 = 216
\]
3. **Multiply these two results**:
\[
6^8 \cdot 6^3 = 1,679,616 \cdot 216
\]
To calculate the multiplication:
\[
1,679,616 \cdot 216 = 362,797,056
\]
Now, if we calculate \(6^{11}\) directly:
\[
6^{11} = 6^{8 + 3} = 6^8 \cdot 6^3
\]
Calculating \(6^{11}\) directly:
\[
6^{11} = 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 = 362,797,056
\]
As expected, both methods give the same result:
\[
6^8 \cdot 6^3 = 6^{11} = 362,797,056.
\]
Thus, whether you simplify \(6^8 \cdot 6^3\) by using the Product Rule or by calculating each exponent directly, you arrive at the same final numerical result of \(362,797,056\).
1. **Calculate \(6^8\)**:
\[
6^8 = 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 = 1,679,616
\]
2. **Calculate \(6^3\)**:
\[
6^3 = 6 \times 6 \times 6 = 216
\]
3. **Multiply these two results**:
\[
6^8 \cdot 6^3 = 1,679,616 \cdot 216
\]
To calculate the multiplication:
\[
1,679,616 \cdot 216 = 362,797,056
\]
Now, if we calculate \(6^{11}\) directly:
\[
6^{11} = 6^{8 + 3} = 6^8 \cdot 6^3
\]
Calculating \(6^{11}\) directly:
\[
6^{11} = 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6 = 362,797,056
\]
As expected, both methods give the same result:
\[
6^8 \cdot 6^3 = 6^{11} = 362,797,056.
\]
Thus, whether you simplify \(6^8 \cdot 6^3\) by using the Product Rule or by calculating each exponent directly, you arrive at the same final numerical result of \(362,797,056\).
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