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Chapter 0: Problem 124

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

Short Answer

Expert verified
Using the product rule for exponents, which states that for any numbers a, n, and m, the equation \(a^{n} \cdot a^{m} = a^{n+m}\) is true, we get \(2^{3} \cdot 2^{5} = 2^{8}\).

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, the equation \(a^{n} \cdot a^{m} = a^{n+m}\) holds true. This means that when multiplying two powers (like \(a^{n}\) and \(a^{m}\)) that share the same base (in this case a), their exponents can be added together.
02

Applying the Product Rule

Now apply the product rule to the specific example given, \(2^{3} \cdot 2^{5}\). Here, the base 'a' is 2, 'n' is 3 and 'm' is 5. According to the product rule, add these two exponents together: \(2^{3} \cdot 2^{5} = 2^{3+5}\).
03

Simplifying the Equation

The final step is to simplify the equation \(2^{3+5}\) by performing the operation in the exponent. This gives: \(2^{8}\). Thus, \(2^{3} \cdot 2^{5} = 2^{8}\).

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Most popular questions from this chapter

Explain the quotient rule for exponents. Use \(\frac{5^{8}}{5^{2}}\) in your explanation.

Find all integers b so that the trinomial can be factored. $$ x^{2}+b x+15 $$

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