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Chapter 5: Problem 262

Find each product. $$ \left(4 a^{3} b\right)\left(9 a^{2} b^{6}\right) $$

Short Answer

Expert verified
36a^5b^7

Step by step solution

01

- Rearrange and Group Like Terms

Write the expression by grouping the like terms together. \[(4a^3b)(9a^2b^6) = 4 \times 9 \times a^3 \times a^2 \times b \times b^6 \]
02

- Multiply the Coefficients

Multiply the numerical coefficients. \[4 \times 9 = 36\]
03

- Apply the Product of Powers Property for 'a'

Combine the like terms involving 'a' using the property \(a^m \times a^n = a^{m+n}\). \[ a^3 \times a^2 = a^{3+2} = a^5\]
04

- Apply the Product of Powers Property for 'b'

Combine the like terms involving 'b' using the property \(b^m \times b^n = b^{m+n}\). \[b \times b^6 = b^{1+6} = b^7\]
05

- Combine All Multipliers

Put together all the multiplied terms to get the final product. \[36a^5b^7\]

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Key Concepts

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

Product of Powers
When multiplying terms with the same base, we apply the product of powers property. This helps simplify expressions by combining exponents. For example, in the expression \(a^3 \times a^2\), both terms have the base \(a\).
According to the product of powers property: \[a^m \times a^n = a^{m+n} \] Here, \(m = 3\) and \(n = 2\), so \(a^3 \times a^2 = a^{3+2} = a^5\).
This principle applies to any base, whether it's a variable or a number. Remember, when the bases are the same, you just add the exponents.
Applied to our exercise:
\[ b \times b^6 = b^{1+6} = b^7 \] The property effectively simplifies the multiplication, helping us manage numerous and potentially complex terms.
Combining Like Terms
Combining like terms means grouping together terms that have the same variable raised to the same power. This process helps simplify complex expressions and equations.
For example, consider \[ \text{4a}^3\text{b} \times \text{9a}^2\text{b}^6 \] Rearrange by grouping the like terms together:
\[ 4 \times 9 \times a^3 \times a^2 \times b \times b^6 \]
Each set of variables can now be simplified:
- \( a^3 \times a^2 \)
- \( b \times b^6 \)
Combining terms with the same variable reduces the complexity of the expression and makes subsequent operations much easier.
Coefficients Multiplication
Multiplying coefficients means multiplying the numbers (constants) that appear before the variables in a term.
In the given example, we have: \[ 4a^3b \times 9a^2b^6 \]
First, let's focus on the coefficients, 4, and 9. Multiply these to get:
\[ 4 \times 9 = 36 \] This is straightforward and simplifies our overall expression, enabling us to easily proceed with combining like terms for the variables:
\[ 4 \times 9 \times a^3 \times a^2 \times b \times b^6 = 36 \times a^5 \times b^7 \] By handling the multiplication of coefficients first, you simplify the problem, making it easier to keep track of all parts of the expression.

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