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Chapter 9: Problem 14

Factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ b^{3 x}-(2 b)^{-1+2 x} $$

Short Answer

Expert verified
Factored form: \(b^{x}.\(b^{2x}-b^{-x}/2)\)

Step by step solution

01

Factor Out

Start by factoring out \(b^{x}\) from each term in the expression: \(b^{3x} - (2b)^{-1+2x}\) can be rewritten as \(b^{x} \cdot b^{2x} - b^{x} \cdot (2^{-1}b^{x})\)
02

Simplify

Simplify the resulting expression: \(b^{x} (b^{2x} - 2^{-1}b^{x})\) = \(b^{x}(b^{2x}-b^{-x}/2)\)
03

Check the Solution

Check if the factored form is correct by multiplying back the expressions: \(b^{x}(b^{2x}-b^{-x}/2)\) multiplied back gives \(b^{3x}-b^{x}/2\) which is simpliefied to \(b^{3x} - (2b)^{-1+2x}\), which is the original equation, confirming the factoring is correct

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Key Concepts

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

Exponent Rules
Mastering exponent rules is crucial when dealing with algebraic expressions involving powers. A solid understanding of these rules facilitates the simplification of complex expressions and plays a major role in various areas of mathematics.

Basic exponent rules include:
  • Product of Powers: When multiplying powers with the same base, you add the exponents, such as in \( b^m \times b^n = b^{m+n} \).
  • Quotient of Powers: When dividing powers with the same base, you subtract the exponents, like \( b^m \div b^n = b^{m-n} \).
  • Power of a Power: When raising a power to another power, you multiply the exponents, demonstrated as \( (b^m)^n = b^{m \times n} \).
  • Negative Exponent: Indicates reciprocal, which means \( b^{-n} = 1\/b^n \).
Apply these rules correctly, and you can transform expressions to more manageable forms or factor common terms as seen in the provided exercise. It's the manipulation of exponents according to these rules that enables the factoring and simplifying process.
Factoring Out Common Factors
Factoring out common factors is a technique used to simplify expressions by identifying and removing factors common to all terms. This core concept is often a first step in solving complex equations and simplifying algebraic expressions.

To factor out a common factor:
  • Identify the smallest exponent for the common base in the terms.
  • Use this as your common factor to remove from each term.
  • Divide each term by this common factor and write it separated from the remaining expression.
In our exercise, we note that \(b^x\) is the common factor in each term in the expression \(b^{3x} - (2b)^{-1+2x}\). By using the smallest exponent of the variable \(b\), which is \(x\), we factor out \(b^x\) from each term, simplifying the process of manipulation that follows.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is all about expressing equations in their simplest form, making it easier to understand and solve them. This process usually involves factoring, combining like terms, and applying exponent rules.

For simplification, follow these guidelines:
  • Combine like terms, which are terms with the same variable raised to the same power.
  • Reduce fractions and simplify any numerical coefficients.
  • Apply exponent rules to simplify terms with variables raised to powers.
As shown in the solution steps, after factoring out \(b^x\), the expression is simplified by applying the exponent and coefficient rules, resulting in \(b^x(b^{2x}-b^{-x}/2)\). Simplifying algebraic expressions not only leads to more concise solutions but also makes it possible to more easily identify further steps necessary for fully solving algebraic problems.

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Most popular questions from this chapter

Devaluation: Due to inflation, a dollar loses its purchasing power with time. Suppose that the dollar loses its purchasing power at a rate of \(2 \%\) per year. (a) Find a formula that gives us the purchasing power of \(\$ 1 t\) years from now. (b) Use your calculator to approximate the number of years it will take for the purchasing power of the dollar to be cut in half.

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

Suppose that in a certain scratch-ticket lottery game, the probability of winning with the purchase of one card is 1 in 500 , or \(0.2 \%\); hence, the probability of losing is \(100 \%-0.2 \%=99.8 \%\). But what if you buy more than one ticket? One way to calculate the probability that you will win at least once if you buy \(n\) tickets is to subtract from \(100 \%\) the probability that you will lose on all \(n\) cards. This is an easy calculation; the probability that you will lose two times in a row is \((99.8 \%)(99.8 \%)=\) (a) What is the probability that you will win at least once if you play three times? (b) Find a formula for \(P(n)\), the percentage chance of winning at least once if you play the game \(n\) times. (c) How many tickets must you buy in order to have a \(25 \%\) chance of winning? A \(50 \%\) chance? (d) Does doubling the number of tickets you buy also double your chances of winning? (e) Sketch a graph of \(P(n) .\) Use \([0,100]\) as the range of the graph. Explain the practical significance of any asymptotes.

Let \(g(t)=3^{5 t} .\) Show that $$ \frac{g(t+h)-g(t)}{h}=g(t) \cdot \frac{g(h)-g(0)}{h} . $$

In the middle of the \(1994-95\) academic year, in the middle of the week, in the middle of the day, there was a bank robbery and subsequent shootout in the middle of Harvard Square. Throughout the afternoon the news spread by word-of- mouth. Suppose that at the time of the occurrence 30 people know the story. Every 15 minutes each person who knows the news passes it along to one other person. Let \(N(t)\) be the number of people who know at time \(t\). (a) Make a table with time in one column and \(N(t)\) in the other. Identify the pattern and write \(N\) as a function of \(t\). (b) If you've written your equation for \(N(t)\) with \(t\) in minutes, convert to hours. If you've done it in hours, convert to minutes. Make a table to check your answers. (It's easy to make a mistake the first time you do this.)
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