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Chapter 0: Problem 122

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

Short Answer

Expert verified
By applying the product rule for exponents to the expression \(2^3 \cdot 2^5\), we get \(2^{(3+5)}\), which simplifies to 256.

Step by step solution

01

Understanding the Product Rule for Exponents

The product rule for exponents states that if you are multiplying two bases that are the same, you can simplify them by keeping the base and adding the exponents. In mathematical terms, this can be expressed as \(a^n \cdot a^m = a^{(n+m)}\) where \(a\) is the base and \(n\) and \(m\) are the exponents.
02

Applying the Product Rule

In the given example, the base is \2\ and the exponents are \3\ and \5\. Using the product rule, the problem simplifies to \(2^3 \cdot 2^5 = 2^{(3+5)}\).
03

Solving the Expression

Adding the exponents, we get \(2^{(8)}\). Evaluating this, i.e., multiplying the base 2 by itself eight times, the final result is 256.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(a x^{2}+c=0, a \neq 0,\) cannot be solved by the quadratic formula.

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Explain how to simplify a rational expression.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}-25}{x-5}=x-5$$

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