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Chapter 4: Problem 14

Apply the product rule for exponents, if possible. $$ 12 \cdot 12^{6} $$

Short Answer

Expert verified
The simplified expression is \(12^7\).

Step by step solution

01

Identify the base and exponents

Recognize that both terms in the product have the same base, which is 12. The first term can be rewritten with an exponent of 1: \(12^1 \times 12^6\).
02

Apply the product rule

Use the product rule for exponents, which states that when multiplying like bases, add the exponents: \(12^1 \times 12^6 = 12^{1+6}\).
03

Simplify the expression

Simplify the exponent by adding the exponents: \(1 + 6 = 7\). Thus, the expression becomes \(12^7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base and Exponents
In mathematics, when dealing with exponents, it is crucial to understand the terms base and exponent. The base is the number that is being multiplied by itself, whereas the exponent tells us how many times the base is used as a factor.
For example, in the expression \(12^6\), 12 is the base and 6 is the exponent.
This simply means that 12 is multiplied by itself 6 times: \(12 \times 12 \times 12 \times 12 \times 12 \times 12\).
Recognizing the base and exponent helps in simplifying complex expressions and applying rules like the product rule efficiently.
Simplifying Exponents
Simplifying exponents often involves using the rules of exponents to make expressions easier to work with. In our example, we started with the expression \(12 \times 12^6\). We first rewrote the lone 12 as \(12^1\) because any number to the power of 1 is itself. Therefore, \(12 \times 12^6\) becomes \(12^1 \times 12^6\).
This type of simplification plays a vital role when applying the product rule or other exponent rules. It makes it easier to see the relationship between the terms and helps prevent errors in calculations.
Multiplying Like Bases
When you multiply terms with the same base, you can use the product rule for exponents. The product rule states that when you multiply like bases, you add the exponents.
For instance, in our exercise with \(12^1 \times 12^6\), we recognized that the base (12) is the same and then added the exponents: \(1 + 6\).
This gives us the new exponent of 7, making the expression \(12^7\). This rule simplifies more complex multiplication problems into manageable steps, provided the bases are the same. Always remember: only the exponents are added, the base remains unchanged.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ 3^{5} \cdot 3^{-6} $$

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Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{3 z}{4}\right)^{-3} $$
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