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Chapter 5: Problem 260

Find each product. $$ \left(\frac{1}{5} f^{8}\right)\left(20 f^{3}\right) $$

Short Answer

Expert verified
The product is \( 4f^{11} \)

Step by step solution

01

Multiply the Constants

First, identify and multiply the constants in the expression. We have \(\frac{1}{5}\) and 20. Multiplying these together: \(\frac{1}{5} \times 20 = 4\)
02

Multiply the Powers of f

Next, multiply the powers of \( f \) by adding their exponents. We have \( f^8 \) and \( f^3 \). Add the exponents: \( 8 + 3 = 11 \)
03

Combine the Results

Combine the results from steps 1 and 2. The product of the constants is 4 and the combined exponent is 11, so the final answer is \( 4f^{11} \)

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Key Concepts

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

Constants Multiplication
Multiplying constants is just like regular multiplication of numbers. Constants are simply the numerical values in your expression, without any variables. In our given exercise, the constants are \(\frac{1}{5}\) and 20.
We start by multiplying these constants together. Think of \(\frac{1}{5}\) as dividing by 5. So, when you multiply \(\frac{1}{5} \times 20\), it is the same as dividing 20 by 5.
Therefore, \(\frac{1}{5} \times 20 = 4\).
So, the product of the constants is 4.
Adding Exponents
Adding exponents happens when you multiply powers of the same base. In our exercise, the base is \f\. We have \f^8\ and \f^3\.
When multiplying these powers, you add their exponents together. Here鈥檚 the rule:
\[ a^m \times a^n = a^{m+n} \]
Applying this rule, we get:
\[ f^8 \times f^3 = f^{8+3} = f^{11} \]
So, the new exponent for \f\ is 11.
Combining Terms
Combining terms involves putting together our results from the previous steps.
In our exercise, we already worked out:
  • The product of the constants is 4
  • The new exponent of \f\ is 11
Now, we combine these to form the final answer.
Hence,
\[ \frac{1}{5} f^8 \times 20 f^3 = 4 f^{11} \]
So, the product of \left( \frac{1}{5} f^8 \right) \times \left( 20 f^3 \right)\ is \4 f^{11}\.

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