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Chapter 3: Problem 3

Each expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{8 x^{9}}{3 x^{7}}$$

Short Answer

Expert verified
The simplified expression is \(\frac{8}{3}x^2\).

Step by step solution

01

Simplify the expression

Start by writing the expression as it is: \( \frac{8x^9}{3x^7} \). The goal is to simplify the fraction by using the properties of exponents and division.
02

Simplify the coefficients

First, simplify the coefficients. In this case, 8 and 3 cannot be simplified further as they do not have common factors. Hence, keep them as they are: \( \frac{8}{3} \).
03

Apply the properties of exponents

When dividing expressions with the same base, you subtract the exponents: \( x^9 \div x^7 = x^{9-7} = x^2 \).
04

Combine simplified components

Combine the simplified coefficient and the simplified base with the new exponent: \( \frac{8}{3}x^2 \).
05

Review the final result

The simplified form of the expression \( \frac{8x^9}{3x^7} \) is \( \frac{8}{3}x^2 \).

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Key Concepts

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

Properties of Exponents
Understanding the properties of exponents can make algebraic expressions less daunting. One of the key properties is that when you divide two powers with the same base, you simply subtract their exponents. For example, in the expression \( x^9 \div x^7 \), both numbers have the base \( x \). According to the property, the operation becomes \( x^{9-7} = x^2 \).
This property allows you to simplify expressions without evaluating large powers directly, which can be quite tedious. Remember that the base remains the same while you only manipulate the exponents.
  • Always check if the bases are identical.
  • Subtract the exponent of the denominator from the exponent of the numerator.
  • Simplification results in a new power expression with the same base.
Mastering this property will immensely help you handle more complex algebraic expressions and equations.
Expression Simplification
Simplifying expressions means making them as basic as possible. This involves reducing any fractions and applying mathematical properties such as exponents simplification.
Start by identifying if there are any coefficients that can be simplified. For instance, in our original expression \( \frac{8x^9}{3x^7} \), we first look at the coefficients 8 and 3. Since they share no common factors, \( \frac{8}{3} \) remains as it is.
Next, apply the properties of exponents to simplify any power terms. In our example, we derived \( x^{9-7} = x^2 \).
  • Check each component separately: coefficients and variables.
  • Be attentive to signs, especially with subtraction in exponents.
  • Combine the simplified parts to get the final reduced expression.
Mastering expression simplification allows you to work with much easier and manageable forms of math problems.
Fraction Operations
Fraction operations involve both the manipulation of numerical coefficients and variables with exponents. The rules for fraction operations you learn early on still apply: if you multiply, divide, add, or subtract fractions, you must have common efforts or bases.
In the expression \( \frac{8x^9}{3x^7} \), our goal first was to simplify. Steps included checking if the coefficients could be reduced, and subsequently, simplifying using exponent properties for the variable part.
When dealing with complex fractions:
  • Always look for common factors to simplify the coefficients.
  • Apply properties of exponents to simplify variable parts.
  • Recheck your work by ensuring your new fraction is in its simplest terms.
Overall, proficiency with fraction operations is an essential mathematical skill that helps with Algebra, Calculus, and beyond.

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Most popular questions from this chapter

Use the given zero to completely factor \(P(x)\) into linear factors. $$\text { Zero: } 5 i ; \quad P(x)=x^{4}-x^{3}+23 x^{2}-25 x-50$$

Divide. $$\frac{8 x^{3}+10 x^{2}-12 x-15}{2 x^{2}-3}$$

Solve each problem. Air Density As the altitude increases, air becomes thinner, or less dense. An approximation of the density \(d\) of air at an altitude of \(x\) meters above sea level is $$d(x)=\left(3.32 \times 10^{-9}\right) x^{2}-\left(1.14 \times 10^{-4}\right) x+1.22$$ The output is the density of air in kilograms per cubic meter. The domain of \(d\) is \(0 \leq x \leq 10,000 .\) (Source: A. Miller and J. Thompson, Elements of Meteorology.) (a) Denver is sometimes referred to as the mile-high city. Compare the density of air at sea level and in Denver. (Hint: \(1 \mathrm{ft} \approx 0.305 \mathrm{m}\) ) (b) Determine the altitudes where the density is greater than 1 kilogram per cubic meter.

Find all rational zeros of each polynomial function. $$P(x)=\frac{1}{6} x^{4}-\frac{11}{12} x^{3}+\frac{7}{6} x^{2}-\frac{11}{12} x+1$$

For each polynomial function, do the following in order. (a) Use Descartes' rule of signs to find the possible number of positive and negative real zeros. (b) Use the rational zeros theorem to determine the possible rational zeros of the function. (c) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other nonreal complex zeros, if any. (f) Find the \(x\) -intercepts of the graph, if any. (g) Find the \(y\) -intercept of the graph. (h) Use synthetic division to find \(P(4),\) and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (i) Sketch the graph. (You may wish to support your answer with a calculator graph.) $$P(x)=3 x^{4}-14 x^{2}-5$$
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