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Chapter 10: Problem 286

In the following exercises, divide the monomials. $$\frac{24 p^{7} q}{6 p^{2} q^{5}}$$

Short Answer

Expert verified
The simplified form is \( \frac{4 p^5}{q^4} \).

Step by step solution

01

- Simplify the Coefficients

First, simplify the coefficients by dividing the numbers. Divide 24 by 6: 24 梅 6 = 4. So, the simplified coefficient is 4.
02

- Simplify the Variable 'p'

Next, simplify the variables with base 'p'. Use the rule \( \frac{a^m}{a^n} = a^{m-n} \): \( \frac{p^7}{p^2} = p^{7-2} = p^5 \).
03

- Simplify the Variable 'q'

Now, simplify the variables with base 'q'. Use the same rule \( \frac{a^m}{a^n} = a^{m-n} \): \( \frac{q}{q^5} = q^{1-5} = q^{-4} = \frac{1}{q^4} \).
04

- Combine Simplified Parts

Combine all the simplified parts: \( 4 p^5 \frac{1}{q^4} \). This can be rewritten as: \( \frac{4 p^5}{q^4} \).

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Key Concepts

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

Simplifying Coefficients
When dividing monomials, it's crucial to first simplify the coefficients. Coefficients are the numerical parts of the terms. In our example, we have the coefficients 24 and 6. To simplify them, divide 24 by 6. This gives us 4. So, the simplified coefficient is 4. Always handle the coefficients first because this step is typically straightforward and sets a clear path for variable simplification.
Exponent Rules
Next, we need to simplify the variables by using exponent rules. A key rule for dividing exponents is \( \frac{a^m}{a^n} = a^{m-n} \). This means that when you divide powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. In our exercise, we have variables with bases 'p' and 'q'.
Variable Simplification
First, let鈥檚 handle the variable 'p'. We apply the exponent rule: \( \frac{p^7}{p^2} = p^{7-2} = p^5 \). Now for 'q': \( \frac{q}{q^5} \) can be simplified as \( q^{1-5} = q^{-4} \). Since \( q^{-4} \) means \( \frac{1}{q^4} \), we can rewrite it as such. Clearly separating these steps can help avoid confusion and ensures each variable is simplified correctly.
Combined Expression
Finally, we combine all the simplified parts. We have a coefficient of 4, \( p^5 \), and \( \frac{1}{q^4} \). Combining these, our final expression becomes \( 4p^5 \frac{1}{q^4} \). To make it clearer, we can write this as a single fraction: \( \frac{4 p^5}{q^4} \). Reviewing each step ensures the solution is both accurate and understandable. This combined expression is neat and simplified for further use or analysis.

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Most popular questions from this chapter

Simplify. $$5 \cdot 2^{-4}$$

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Divide the monomials. $$\frac{\left(5 m^{9} n^{3}\right)\left(8 m^{3} n^{2}\right)}{\left(10 m n^{4}\right)\left(m^{2} n^{5}\right)}$$

Simplify each expression. $$(s+8)(s+9)$$

Multiply the binomials using various methods. $$(5 u-3)(u+8)$$
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