Problem 409 Divide the monomials. $$-72 u^... [FREE SOLUTION]

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Chapter 6: Problem 409

Divide the monomials. $$-72 u^{12} \div 12 u^{4}$$

Short Answer

Expert verified

-6u^8

Step by step solution

Divide the coefficients

The coefficients are -72 (numerator) and 12 (denominator). Divide -72 by 12. -72 梅 12 = -6

Use the quotient rule for exponents

When dividing like bases, subtract the exponent of the denominator from the exponent of the numerator. The exponents are 12 (numerator) and 4 (denominator). Apply the rule: $u^{12 - 4} = u^8$

Combine the results

Combine the results from Step 1 and Step 2. The division of the monomials is: -6u^8

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Key Concepts

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

monomials

A monomial is a type of algebraic expression. It has only one term, which includes a coefficient and variables raised to non-negative integer exponents.

Examples include:

3x
-5y^3
7ab^2

The term '$-72u^{12}$' from the exercise is a monomial because it consists of a single term with a coefficient (-72) and a variable (u) raised to an exponent (12). Understanding monomials is crucial for algebraic operations like addition, subtraction, multiplication, and division.

quotient rule

The quotient rule is an essential concept in algebra, especially when dealing with exponents. It helps to simplify fractions involving like bases. The rule states: \[ \frac{a^m}{a^n} = a^{m-n} \] In simpler terms, when dividing two like bases, subtract the exponent in the denominator from the exponent in the numerator.

In our exercise, we have the variable 'u' with exponents 12 and 4:
\[ u^{12} \times u^{-4} = u^{12-4} = u^8 \]
This makes the handling of large exponent values straightforward and helps in reducing expression complexity.

exponents

Exponents show how many times a number, known as the base, is multiplied by itself. For instance, in the expression '$u^4$', the base is '$u$' and the exponent is '$4$', indicating: \[ u \times u \times u \times u \]

Some important rules involving exponents include:

Product Rule: \[ a^m \times a^n = a^{m+n} \]
Power Rule: \[ (a^m)^n = a^{mn} \]
Quotient Rule: \[ \frac{a^m}{a^n} = a^{m-n} \]

Mastering these rules is essential to maneuver through complex algebraic problems effortlessly.

coefficients

The coefficient is the numerical part of a term that contains both numbers and variables. In the expression '$5x$', '5' is the coefficient. For our exercise, the coefficients of the two monomials are -72 and 12.

When dividing monomials, first divide their coefficients separately. In our specific example:
\[ \frac{-72}{12} = -6 \]
This process simplifies the division task and makes managing the variable part easier. Handling coefficients correctly ensures accurate results in algebraic equations.

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91影视

Divide the monomials. $$-72 u^{12} \div 12 u^{4}$$

Short Answer

Step by step solution

Divide the coefficients

Use the quotient rule for exponents

Combine the results

Key Concepts

monomials

quotient rule

exponents

coefficients

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