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Chapter 12: Problem 113

Simplify. $$ a^{12} \cdot a^{6} $$

Short Answer

Expert verified
The simplified expression is \( a^{18} \).

Step by step solution

01

Identify the properties of exponents

To simplify the expression, use the property of exponents which states that when multiplying two powers with the same base, you add the exponents: \[ a^m \times a^n = a^{m+n} \]
02

Apply the property of exponents

Given the expression \( a^{12} \times a^6 \), apply the property of exponents: \[ a^{12+6} \]
03

Simplify the exponent

Add the exponents: \(12 + 6 = 18\), so the expression becomes: \[ a^{18} \]

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Key Concepts

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

properties of exponents
Exponents are a special way of representing repeated multiplication. They have rules that make working with them easier. One important rule is the Product of Powers Property.
This property states that when you multiply two powers with the same base, you simply add their exponents. For example:
  • When you see something like \(a^m \times a^n\), you can combine the exponents: \(a^m \times a^n = a^{m+n}\)
  • This means that if you have \(2^3 \times 2^4\), it equals \(2^{3+4} = 2^7\).

Understanding this property can help you simplify expressions quickly and correctly. It's a cornerstone of working with exponents in algebra.
multiplying powers
When you come across multiplication of exponents in algebraic expressions, you need to follow the principles we discussed. If the bases are the same, you consolidate them by adding up the exponents.
Let's take a closer look with an example: \(a^{12} \times a^6\).
  • Step 1: Identify the property involved. Here, since we are multiplying powers of the same base, we use the Product of Powers Property.
  • Step 2: Apply the property. We add the exponents: \(a^{12+6}\).
  • Step 3: Calculate the new exponent. \(12 + 6 = 18\).
So, \(a^{12} \times a^6\) simplifies to \(a^{18}\).
Multiplying powers can look tricky at first, but using the properties of exponents makes it much easier!
algebraic expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. When dealing with exponents, it's important to understand how they fit into algebraic expressions.
Here are some key points to remember:
  • Variables like \(a\) can have exponents, which tell you how many times to multiply the variable by itself.
  • Exponents follow specific rules that help you simplify expressions involving them, like the Product of Powers Property.
  • Simplifying algebraic expressions often involves combining like terms, which includes adding or subtracting exponents correctly.
For example, in the expression \(a^{12} \times a^6\), we simplify it by recognizing the common base and applying the exponent rules we discussed.
Carefully following the steps ensures you get the right answer, making working with algebraic expressions more straightforward.

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