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Chapter 5: Problem 12

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

Short Answer

Expert verified
12^7

Step by step solution

01

Identify the base and exponents

Recognize that both terms have the same base, 12. The two terms are 12 (which can be written as 12^1) and 12^6.
02

Apply the product rule for exponents

The product rule for exponents states that when multiplying like bases, the exponents are added. Specifically, for terms \(a^m \times a^n = a^{m+n}\). In this case, it applies as follows: 12^1 \times 12^6 = 12^{1+6}.
03

Solve the exponent

Add the exponents: 1 + 6 = 7 Therefore, the expression simplifies to 12^7.

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental math concept used to express repeated multiplication of a number by itself. For example, instead of writing 2 * 2 * 2, we use an exponent: 2鲁. Here, 2 is the base, and 3 is the exponent. The value of 2鲁 is 8. Exponents make it simpler to represent large or repetitive multiplication.

An exponent indicates how many times the base is multiplied by itself. In the expression 5虏, 5 is the base, and 2 is the exponent, representing 5 * 5, which equals 25.
Multiplying Like Bases
When multiplying numbers with the same base, we can use the product rule for exponents. This rule states that you should add the exponents together. Here鈥檚 the product rule formula:

\[a^m \times a^n = a^{m+n}\]
For example, if we have 3虏 * 3鲁, both terms have the base 3. Using the product rule, we add the exponents: 2 + 3, giving us 3鈦�, which equals 243.

Let鈥檚 look at our original problem: \[12 \times 12^6\]. First, note that 12 can be written as \[12^1\]. Now, both terms have the same base, 12. Applying the product rule, we add the exponents: \[1 + 6 = 7\]. So the expression simplifies to \[12^7\].
Simplifying Expressions
Simplifying expressions involves reducing them to their most basic form. This makes mathematical operations easier to handle. One common step in simplifying expressions with exponents is using the product rule. Let's break down our exercise further.

Initially, we had \[12 \times 12^6\].
By recognizing that \[12 = 12^1\], both terms now have the same base and fit the product rule for exponents. Adding the exponents: \[1 + 6\] gives us \[12^7\]. This is the simplest form of the expression.

Why simplify? It helps in performing further calculations and in understanding the relationships between terms more clearly. Simplification is a crucial skill in algebra, calculus, and beyond.

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