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Chapter 10: Problem 12

Multiply.$$b \cdot 7 b^{10} \cdot 5 b^{8}$$

Short Answer

Expert verified
The expression simplifies to \( 35b^{19} \).

Step by step solution

01

Rearrange the Terms

The expression to be multiplied is \( b \cdot 7b^{10} \cdot 5b^{8} \). First, let's rearrange the terms, bringing all coefficients together and all powers of \( b \) together: \( (7 \cdot 5) \cdot (b \cdot b^{10} \cdot b^{8}) \).
02

Multiply the Coefficients

The coefficients in the expression are 7 and 5. Multiply these numbers together: \( 7 \cdot 5 = 35 \). The expression now looks like: \( 35 \cdot (b \cdot b^{10} \cdot b^{8}) \).
03

Simplify the Powers of b

When multiplying like bases, such as \( b \), you add the exponents. Here, you have \( b = b^1 \), \( b^{10} \), and \( b^{8} \). Add the exponents: \( 1 + 10 + 8 = 19 \). So, \( b \cdot b^{10} \cdot b^{8} = b^{19} \).
04

Combine the Results

Now that we've simplified both the coefficients and the powers, combine them into a single expression. The final result is: \( 35b^{19} \).

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Key Concepts

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

Coefficient Multiplication
When multiplying monomials, one of the key steps is coefficient multiplication. Coefficients are the numerical factors in a term. In our exercise, the coefficients are 7 and 5 from the terms \( 7b^{10} \) and \( 5b^{8} \). To multiply coefficients, simply perform ordinary multiplication.
  • First, identify the coefficients: Here, the coefficients are 7 and 5.
  • Next, multiply them: \( 7 \times 5 \).
  • This multiplication results in 35.
So, the numerical factor of the final expression becomes 35. This is an essential building block for simplifying the rest of the expression.
Exponent Rules
Dealing with exponents can be tricky, but understanding the rules makes it easier. The most important rule for this exercise is when multiplying terms with the same base; you add the exponents. Our base is \( b \), and we have \( b \), \( b^{10} \), and \( b^{8} \).
  • Firstly, recognize \( b \) can be written as \( b^1 \) for clarity.
  • The exponents in this problem are 1, 10, and 8.
  • Add these exponents together: \( 1 + 10 + 8 = 19 \).
By applying the exponent addition rule, you simplify the expression to \( b^{19} \). This rule is invaluable when dealing with powers in algebra, helping to keep complex expressions manageable.
Rearranging Terms
Rearranging terms is a critical step to simplify multiplication of monomials. This makes the expression easier to handle. The exercise involves terms \( b \cdot 7b^{10} \cdot 5b^{8} \).
  • Initially, organize the coefficients together: \( (7 \times 5) \).
  • Then gather similar bases: \( (b \cdot b^{10} \cdot b^{8}) \).
This arrangement of like terms ensures all coefficients are multiplied separately, and powers of the same base are combined efficiently.
Placing the coefficients at the beginning solidifies the numerical multiplication, and collecting like bases sets up the exponentiation for simplification. This step is the foundation for clear and correct calculations.

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