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Chapter 5: Problem 105

Explain the product rule for exponents. Use \(2^{3} \cdot 2^{5}\) in your explanation.

Short Answer

Expert verified
The Product Rule for Exponents states that when multiplying two powers that have the same base, you can add the exponents. Therefore, \(2^{3} \cdot 2^{5}\) simplifies to \(2^{8}\) according to the product rule for exponents.

Step by step solution

01

Understanding the Product Rule for Exponents

The product rule for exponents states that for any numbers a, n, and m: \(a^{n} \cdot a^{m} = a^{n+m}\). When multiplying two powers that have the same base, you can add the exponents.
02

Apply the Product Rule to the Given Expression

Apply the product rule to the given expression \(2^{3} \cdot 2^{5}\) by adding the exponents of the two terms. This gives \(2^{3+5}\) as the resulting expression.
03

Simplify the Resulting Expression

Simplify the resulting expression from previous step, \(2^{3+5}\), by adding together the numbers in the exponent to get \(2^{8}\).

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