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Chapter 3: Problem 32

Rewrite each expression as simply as you can. $$4 a^{4} \cdot 3 a^{3}$$

Short Answer

Expert verified
The simplified expression is \ 12 a^{7} \.

Step by step solution

01

Apply the Commutative Property of Multiplication

Combine the numerical coefficients (4 and 3) and the variable terms \(a^{4} \text{ and } a^{3}\) by rewriting the expression: \[4 a^{4} \cdot 3 a^{3} = (4 \cdot 3) \cdot (a^{4} \cdot a^{3}).\]
02

Multiply the Coefficients

Multiply the numerical coefficients together: \[4 \cdot 3 = 12.\] Thus, the expression becomes: \[12 \cdot (a^{4} \cdot a^{3}).\]
03

Use the Product of Powers Property

Apply the product of powers property \(a^{m} \cdot a^{n} = a^{m+n}\) to combine the variable terms: \[a^{4} \cdot a^{3} = a^{4+3} = a^{7}.\]
04

Combine All Parts

Combine the simplified parts to get the final expression: \[12 \cdot a^{7} = 12 a^{7}.\]

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Key Concepts

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

commutative property of multiplication
The Commutative Property of Multiplication states that the order in which you multiply numbers does not change the product. For example, if you have two numbers 4 and 3, then 4 multiplied by 3 is the same as 3 multiplied by 4, both equal to 12.
This property can be very handy when you're working with algebraic expressions. If you see an expression like \((4 \cdot a^{4}) \cdot (3 \cdot a^{3})\), you can rearrange it to make the multiplication simpler.
This is done by placing the numerical coefficients together and the variable terms together, resulting in \((4 \cdot 3) \cdot (a^{4} \cdot a^{3})\).
Rewriting in this way makes it easier to see what needs to be multiplied first.
product of powers property
The Product of Powers Property is used when you multiply two expressions with the same base. The rule states that \((a^{m} \cdot a^{n} = a^{m+n})\).
In simple terms, you just add the exponents.
For example, in the expression \(a^{4} \cdot a^{3}\), you can add the exponents 4 and 3 to get 7.
So, \(a^{4} \cdot a^{3} = a^{7}\).
This simplifies many problems because it allows you to combine like terms into a single term, making the expression much shorter and easier to handle.
multiplying coefficients
Multiplying coefficients is one of the simplest steps in simplifying algebraic expressions. Coefficients are the numerical parts of the terms.
For example, in our expression \(4 \cdot a^{4} \cdot 3 \cdot a^{3}\), the coefficients are 4 and 3.
When you multiply them together, \(4 \cdot 3\), you get 12.
This step helps to condense the expression by combining the numerical values into one coefficient.
Once you have multiplied the coefficients, you're only left with multiplying the variables using the product of powers property.
This can greatly simplify and speed up the process of solving the expression!

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Most popular questions from this chapter

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Find the indicated roots without using a calculator. $$ \sqrt{(-3)^{2}} $$
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