Question

Use the Product Rule of Exponents to simplify 6^8 . 6^3

Answer 1

how did you get that answer?

Answer 2

6^8 \cdot 6^3 = 6^{8 + 3} = 6^{11} like in actual number n schtuff

Answer 3

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 \(6^{11}\).

Answer 4

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 \(6^{11}\).

Answer 5

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 \(362,797,056\).