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Chapter 9: Problem 32

Simplify as much as possible: (a) \(\frac{x^{2 y}+x^{y+2}}{x^{y}}\) (b) \(\frac{\frac{\sqrt{x}}{x^{-1 / 2} y}-1}{y-\frac{x^{2}}{y}}\) (c) \(\frac{A^{B+4}-A^{3 B}}{A^{B}\left(A^{2}-A^{B}\right)}\) (d) \(\frac{y^{3 w}-y^{w+4}}{y^{w}\left(y^{w}+y^{2}\right)}\)

Short Answer

Expert verified
(a) \(x^{y} + x^{2}\), (b) \(\frac{y - y^2}{y^3 - x^2}\), (c) \(\frac{A^{4}-A^{2B}}{A^{2}-A^{B}} \) , (d) \(y^{2 w} - y^{4}\)

Step by step solution

01

Simplify expression (a) \(\frac{x^{2 y}+x^{y+2}}{x^{y}}\)

Use the property of exponents where the division of two exponentials with the same base is equivalent to subtracting their exponents. Simplified expression (a) is \(x^{2y-y} + x^{2}\) which becomes \(x^{y} + x^{2}\)
02

Simplify expression (b) \(\frac{\frac{\sqrt{x}}{x^{-1 / 2} y}-1}{y-\frac{x^{2}}{y}}\)

Begin by clearing the fractions in the numerator and denominator, and then simplify the expression. Clearing the fractions, we have \(\frac{\sqrt{x}* y * x^{1/2} - y^2}{y^3- x^2}\). Once simplified, expression (b) becomes \(\frac{y - y^2}{y^3 - x^2}\)
03

Simplify expression (c) \(\frac{A^{B+4}-A^{3 B}}{A^{B}\left(A^{2}-A^{B}\right)}\)

Similar to (a), use the property of exponents and simplify the expression. Simplified expression (c) is \(\frac{A^{4}-A^{2B}}{A^{2}-A^{B}} \)
04

Simplify expression (d) \(\frac{y^{3 w}-y^{w+4}}{y^{w}\left(y^{w}+y^{2}\right)}\)

Apply the same principle of exponentials as in (a) and (c) to simplify the expression. Simplified expression (d) is \(y^{2 w} - y^{4}\)

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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 is essential when simplifying expressions. Here are some crucial properties:
  • Product of Powers: When multiplying two powers with the same base, you add the exponents. For example, if you have\( x^a \times x^b = x^{a+b} \).

  • Quotient of Powers: When dividing two powers with the same base, you subtract the exponents. For instance,\( \frac{x^a}{x^b} = x^{a-b} \).

  • Power of a Power: If a power is raised to another power, multiply the exponents. In other words,\((x^a)^b = x^{a \cdot b}\).

  • Negative Exponent: A negative exponent means you take the reciprocal of the base. This looks like\( x^{-a} = \frac{1}{x^a}\).

These rules can make complicated expressions easier to handle. They allow you to combine and rearrange terms to simplify them. Familiarity with these properties will make dealing with complex expressions, such as those presented in the exercise, much more manageable.
Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging algebraic expressions to simplify them. This often involves applying operations such as addition, subtraction, multiplication, and division in combination with exponent rules. In the exercise, for example, simplifying each expression involves:
  • Cancelling Common Terms: Identify and eliminate terms numerator and denominator to make expressions simpler. This can occur through factoring.

  • Distributing and Factoring: Use distribution to simplify expressions and factoring to break down complex expressions, making it easier to cancel out terms.

  • Simplifying Fractions: Combine like terms and reduce fractions to their simplest form by ensuring there are no common factors between the numerator and denominator.
Understanding the proper steps and sequences in algebraic manipulation can solve high-complexity expressions easily and systematically.
Rational Expressions
Rational expressions are fractions in which the numerator and the denominator are both polynomials. Simplifying rational expressions involves manipulating these algebraic fractions according to specific rules and operations. In the given exercise, you simplify rational expressions by using the following principles:
  • Factorization: Factor the numerator and denominator separately, turning them into simpler components that can be easily compared and reduced.

  • Finding Common Denominators: It's crucial when adding or subtracting rational expressions, akin to dealing with ordinary fractions.

  • Canceling Common Factors: Once factorized, cancel out any identical terms that appear in both the numerator and denominator.

Rational expressions require consistent practice to master, as they are common in algebra and calculus. The key is understanding the fundamental properties and operations of fractions and polynomials, allowing students to simplify complex algebraic expressions efficiently.

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Most popular questions from this chapter

For Problems 1 through 9, simplify the following expressions. $$ \frac{\left(a b^{x}\right)^{-2}}{(a b)^{-x}} $$

The number of bacteria in a certain culture is known to triple every day. Suppose that at noon today there are 200 bacteria. (a) Construct a table of values to find a function that gives the number of bacteria after \(t\) days. (b) Approximately what was the population count at noon yesterday? At noon 4 days ago? (c) From now on, suppose the population at noon today is called \(B_{0}\) rather than being specifically 200 . Find a function that gives the number of bacteria after \(t\) days. (d) Express the number of bacteria as a function of \(w\), where \(w\) is time measured in weeks. (e) How many bacteria will be present at noon one week from today?

For Problems 16 through 21, if the statement is always true, write "True;" if the statement is not always true, produce a counterexample. In these problems, \(a\) and \(b\) are positive constants. $$ \sqrt{a^{2 x}}=a^{x} $$

In the Exploratory Problems you approximated the derivatives of \(2^{x}, 3^{x}\), and \(10^{x}\) for various values of \(x\), and, after looking at your results, you conjectured about the patterns. Now, using the definition of the derivative of \(f\) at \(x=a\), we return to this, focusing on the function \(f(x)=5^{x}\). (a) Using the definition of the derivative of \(f\) at \(x=a\), $$ f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ give an expression for \(f^{\prime}(0)\), the slope of the tangent line to the graph of at \(x=0\). (b) Show that for the function \(f(x)=5^{x}\), the difference quotient, \(\frac{f(x+h)-f(x)}{h}\), is equal to \(f(x) \cdot \frac{f(h)-f(0)}{h}\). (c) Using the definition of derivative, $$ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} $$ conclude that the derivative of \(f(x)=5^{x}\) is $$ f^{\prime}(0) \cdot f(x) $$ Notice that you have now proven that the derivative of \(5^{x}\) is proportional to \(5^{x}\), with the proportionality constant being the slope of the tangent line to \(5^{x}\) at \(x=0\). $$ f^{\prime}(x)=f^{\prime}(0) \cdot f(x) $$ (d) Approximate the slope of the tangent line to \(5^{x}\) at \(x=0\) numerically.

Factor \(b^{x}\) out of each of the following expressions. (a) \(3 b^{x}-b^{2 x}\) (b) \((3 b)^{x}-b^{x+2}\) (c) \(b^{3 x / 2}-b^{2 x-1}\)
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