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Chapter 6: Problem 416

Divide the monomials. $$\frac{64 q^{11} r^{9} s^{3}}{48 q^{6} r^{8} s^{5}}$$

Short Answer

Expert verified
\(\frac{4}{3} q^5 r s^{-2}\)

Step by step solution

01

- Simplify the coefficients

Divide the numerical coefficients: \ \ \(\frac{64}{48} = \frac{4}{3}\)
02

- Subtract the exponents of 'q'

Apply the law of exponents for division: \ \ \(q^{11} / q^{6} = q^{11-6} = q^{5}\)
03

- Subtract the exponents of 'r'

Similarly, for 'r': \ \ \(r^{9} / r^{8} = r^{9-8} = r^{1}\)
04

- Subtract the exponents of 's'

For 's': \ \ \(s^{3} / s^{5} = s^{3-5} = s^{-2}\)
05

- Combine all simplified parts

Combine all the simplified parts together: \ \ \(\frac{4}{3} q^{5} r^{1} s^{-2}\)

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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, the first step is to simplify the coefficients. Coefficients are the numerical parts of each term. In our example, the coefficients are 64 and 48. We need to divide these numbers: \(\frac{64}{48}\). Start by finding the greatest common divisor (GCD), which is 16. Then, divide the numerator and the denominator by their GCD: \(\frac{64 \, / \, 16}{48 \, / \, 16} = \frac{4}{3}\). This leaves us with the simplified coefficient \(\frac{4}{3}\). Always simplify coefficients first since it makes the rest of your calculation easier.
Law of Exponents
The Law of Exponents is a crucial concept when dividing monomials. The main idea is: for any nonzero number 'a' and positive integers m and n, \(a^m / a^n = a^{m-n}\). This rule lets us subtract the exponent in the denominator from the exponent in the numerator.
  • If the exponent in the numerator is larger, the result will have a positive exponent.
  • If the exponent in the denominator is larger, the result will have a negative exponent.
Exponent Subtraction
We use the Law of Exponents for actual exponent subtraction. Take each variable separately:## For 'q': \(q^{11} / q^6\). Using the law: \(q^{11-6} = q^5\) ## For 'r':\(r^{9} / r^{8}\). Using the law: \(r^{9-8} = r^1\) ## For 's': \(s^{3} / s^{5}\). Using the law: \(s^{3-5} = s^{-2}\). Negative exponents will be discussed next.
Negative Exponents
Negative exponents may seem tricky, but they are straightforward once you understand their meaning. A negative exponent indicates the reciprocal of that base raised to the corresponding positive exponent. In simple terms, \(a^{-n} = 1 / a^n\).In our example: \(s^{-2}\) means \(1 / s^2\). When you have a negative exponent, remember:
  • Move the term with the negative exponent to the denominator.
  • Change the exponent to a positive one. This results in our final expression as: \(\frac{4}{3} q^5 r s^{-2} = \frac{4 q^5 r}{3 s^2}\).

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Most popular questions from this chapter

Mixed Practice $$\left(8 x^{5}\right)(9 x) \div 6 x^{3}$$

Simplify. (a) \(2^{-8}\) (b) \(10^{-2}\)

Divide each polynomial by the monomial. $$\frac{51 y^{4}+42 y^{2}}{3 y^{2}}$$

Divide each polynomial by the monomial. $$\left(9 n^{4}+6 n^{3}\right) \div 3 n$$

Mixed Practice (a) \(\frac{z^{6}}{z^{5}}\) (b) \(\frac{z^{5}}{z^{6}}\)
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