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

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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 \(6^{11}\).

how did you get that answer?

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 \(6^{11}\).

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

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 \(362,797,056\).