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Chapter 10: Problem 282

In the following exercises, divide the monomials. $$\frac{36 y^{9}}{4 y^{7}}$$

Short Answer

Expert verified
The result is \( 9 y^2 \).

Step by step solution

01

Identify the coefficients

The coefficients of the monomials are 36 (numerator) and 4 (denominator).
02

Divide the coefficients

Divide the coefficients: \ \ \( \frac{36}{4} = 9 \)
03

Identify the variables and their exponents

The variable in both the numerator and the denominator is \( y \). The exponent of \( y \) in the numerator is 9, and in the denominator, it is 7.
04

Apply the exponent rule for division

Subtract the exponent of \( y \) in the denominator from the exponent of \( y \) in the numerator: \ \ \( y^{9-7} = y^2 \)
05

Combine the results

Combine the divided coefficients and the simplified exponents: \ \ \( \frac{36 y^9}{4 y^7} = 9 y^2 \)

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Key Concepts

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

Coefficients in Algebra
In algebra, coefficients are the numerical part of a term that includes a variable. For example, in the term '36y\textsuperscript{9}', '36' is the coefficient. Coefficients can be positive or negative numbers, and they can be found in all kinds of algebraic expressions.
Identifying coefficients is crucial when dividing monomials. In our exercise, we need to divide the coefficients separately before we handle the variables. This simplifies the process.
Here鈥檚 what we did in the exercise:
  • We identified '36' as the coefficient in the numerator.
  • We identified '4' as the coefficient in the denominator.
These coefficients were then divided:ewline ewline \(\frac{36}{4} = 9\).ewline ewline Thus, the coefficient part of our solution is '9'.
Exponent Rules
Exponent rules are essential when simplifying algebraic expressions that involve powers of variables. These rules allow us to manipulate expressions in a manageable way. One of the most important rules for division is: ewline ewline \[\frac{a^m}{a^n} = a^{m-n} \], where \(a\) is a variable and \(m, n\) are its exponents.
In our exercise, both the numerator and denominator have the variable \(y\):
  • The numerator has \(y^9\).
  • The denominator has \(y^7\).
This means we use the exponent division rule:ewline \(\frac{y^9}{y^7} = y^{9-7} = y^2\).ewline Applying this rule greatly simplifies the expression by subtracting the exponents.
Simplifying Expressions
Simplifying expressions is the process of rewriting them in a more concise and understandable way. For monomials, this often involves dividing coefficients and using exponent rules to handle the variables.
To simplify our example expression, follow these steps:
  • Divide the coefficients: \(\frac{36}{4} = 9\)
  • Apply the exponent rule: \(\frac{y^9}{y^7} = y^{2}\)
Combining these two results gives us:ewline ewline \(\frac{36 y^9}{4 y^7} = 9 y^2\).ewline This is the simplified form of our original expression. Remember, the goal of simplifying is to make an expression as easy to work with as possible.

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