Chapter 2: Problem 7
Divide and simplify. $$31.4 a b^{9} \text { by } 2.66 a b^{8}$$
Short Answer
Expert verified
The simplified result is approximately \(11.8b\).
Step by step solution
01
Identify the Numerator and Denominator
The exercise requires us to divide two algebraic expressions. The numerator is the term we are dividing, which is \(31.4ab^9\), and the denominator is the term by which we are dividing, which is \(2.66ab^8\). Our division expression is \(\frac{31.4ab^9}{2.66ab^8}\).
02
Simplify the Coefficients
Divide the coefficients (the numerical parts) of the terms to simplify the expression. To divide the coefficients, we perform \(\frac{31.4}{2.66}\), which simplifies to approximately 11.8.
03
Cancel the Common Factors in Variables
Since the variables in the expression are the same except for their exponents, cancel out the common variable factor. In this expression, the variable 'a' is present in both the numerator and the denominator, so it cancels out. Similarly, \(b^8\) cancels out as it is a common factor, leaving us with \(b^1\) in the numerator since \(b^9 \div b^8 = b^{9-8} = b^1\).
04
Write the Simplified Expression
Put together the simplified coefficient and the remaining variable factor to get the result of the division: \(11.8b\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Dividing Polynomials
Dividing polynomials is a process similar to numerical division but involves variables with exponents. The goal is to simplify the expression by dividing the terms of the numerator by the terms of the denominator. When performing polynomial division, it's important to divide the coefficients (numerical parts) and the variables separately. If the variables have exponents, you subtract the exponents of the denominator from those of the numerator.
Example: Given the problem \(31.4ab^{9}\) divided by \(2.66ab^{8}\), you would first divide the coefficients 31.4 by 2.66, and then simplify the variable part by subtracting the exponent of b in the denominator from the exponent of b in the numerator, which gives you \(b^{9-8}\). Thus, the solution simplifies to \(11.8b\).
This procedure often involves dealing with fractions, and can be facilitated by knowledge of powers and the laws governing them. Always remember to write your final answer in the simplest form.
Example: Given the problem \(31.4ab^{9}\) divided by \(2.66ab^{8}\), you would first divide the coefficients 31.4 by 2.66, and then simplify the variable part by subtracting the exponent of b in the denominator from the exponent of b in the numerator, which gives you \(b^{9-8}\). Thus, the solution simplifies to \(11.8b\).
This procedure often involves dealing with fractions, and can be facilitated by knowledge of powers and the laws governing them. Always remember to write your final answer in the simplest form.
Simplifying Coefficients
The coefficients in an algebraic expression are the numerical multipliers of the variables. Simplifying coefficients involves performing arithmetic operations like addition, subtraction, multiplication, or division on these numbers.
For instance, when dividing polynomials, the coefficients of like terms can be divided by each other, just as you would divide plain numbers. As seen in our exercise, dividing \(31.4\) by \(2.66\) results in approximately 11.8. An important aspect of simplifying coefficients is to ensure the division is done correctly and expressed as precisely as needed for the problem at hand, whether it be as a decimal, fraction, or whole number.
Understanding decimals and fractions, as well as using a calculator correctly, are key skills for simplifying coefficients accurately when working with algebraic expressions.
For instance, when dividing polynomials, the coefficients of like terms can be divided by each other, just as you would divide plain numbers. As seen in our exercise, dividing \(31.4\) by \(2.66\) results in approximately 11.8. An important aspect of simplifying coefficients is to ensure the division is done correctly and expressed as precisely as needed for the problem at hand, whether it be as a decimal, fraction, or whole number.
Understanding decimals and fractions, as well as using a calculator correctly, are key skills for simplifying coefficients accurately when working with algebraic expressions.
Canceling Common Factors
Canceling common factors in both the numerator and denominator of a fraction is a crucial step in simplification. This method relies on the fundamental principle that a quantity divided by itself equals one. When variables with exponents occur in both numerator and denominator, and they are the same base, you may cancel them by subtracting the lower exponent from the higher one.
For example: In the expression \(\frac{31.4ab^{9}}{2.66ab^{8}}\), the variable 'a' is present in both terms so it cancels out. This is because \(\frac{a}{a} = 1\). Similarly, \(b^8\) is a common factor in both the numerator and the denominator, thus \(b^9 \div b^8 = b^{9-8} = b\), leaving us with a single \(b\) in the numerator. It's critical to recognize common factors to simplify expressions to their lowest terms and avoid making calculation mistakes.
For example: In the expression \(\frac{31.4ab^{9}}{2.66ab^{8}}\), the variable 'a' is present in both terms so it cancels out. This is because \(\frac{a}{a} = 1\). Similarly, \(b^8\) is a common factor in both the numerator and the denominator, thus \(b^9 \div b^8 = b^{9-8} = b\), leaving us with a single \(b\) in the numerator. It's critical to recognize common factors to simplify expressions to their lowest terms and avoid making calculation mistakes.
Algebraic Division
Algebraic division is the process of dividing algebraic expressions that contain variables. It is governed by the same rules as numerical division but additionally involves the laws of exponents when dealing with variables with powers. The division of polynomials, binomials, and monomials all fall under this category and require careful manipulation of both coefficients and variables.
Key strategies for successful algebraic division include simplifying coefficients, canceling common factors, and rewriting subtractive exponents when necessary. The exercise presents a simple form of algebraic division with monomials, showing how a process similar to what we use for numbers is extended to variables and their exponents. Understanding algebraic division helps in later studying more complex concepts such as polynomial long division and synthetic division.
Key strategies for successful algebraic division include simplifying coefficients, canceling common factors, and rewriting subtractive exponents when necessary. The exercise presents a simple form of algebraic division with monomials, showing how a process similar to what we use for numbers is extended to variables and their exponents. Understanding algebraic division helps in later studying more complex concepts such as polynomial long division and synthetic division.
