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Chapter 8: Problem 122

Simplify each expression. Write answers using only positive exponents. $$ \left(\frac{a^{-2} b^{3} a^{5} b^{-2}}{a^{6} b^{-5}}\right)^{-4} $$

Short Answer

Expert verified
The simplified expression is \( \frac{a^{12}}{b^{24}} \).

Step by step solution

01

Simplify the Expression Inside the Parentheses

First, let's simplify the expression inside the parentheses. The given expression is \( \frac{a^{-2} b^{3} a^{5} b^{-2}}{a^{6} b^{-5}} \). Begin by combining the exponents for like bases in the numerator: \( a^{-2} \cdot a^{5} = a^{-2+5} = a^{3} \) and \( b^{3} \cdot b^{-2} = b^{3-2} = b^{1} = b \). This simplifies the numerator to \( a^{3} b^{1} \) or just \( a^{3} b \). So the expression becomes \( \frac{a^{3} b}{a^{6} b^{-5}} \).
02

Apply the Quotient Rule for Exponents

Now let's use the quotient rule \( \frac{x^m}{x^n} = x^{m-n} \) to simplify the fraction: \[ \frac{a^{3} b^{1}}{a^{6} b^{-5}} = a^{3-6} b^{1-(-5)} = a^{-3} b^{6} \]. Now, the expression inside the parentheses is \( a^{-3} b^{6} \).
03

Apply the Negative Exponent

The entire expression is raised to the power of \(-4\) as follows: \( \left(a^{-3} b^{6}\right)^{-4} \). Using the power of a power rule \((x^m)^n = x^{m \cdot n}\), we simplify know that: \[ a^{-3 \cdot (-4)} b^{6 \cdot (-4)} = a^{12} b^{-24} \].
04

Convert Negative Exponents to Positive

Finally, to express the expression with only positive exponents, write \( b^{-24} \) as \( \frac{1}{b^{24}} \): \[ a^{12} b^{-24} = \frac{a^{12}}{b^{24}} \]. Thus, the simplified expression is \( \frac{a^{12}}{b^{24}} \).

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Key Concepts

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

Exponent Rules
Exponent rules are a helpful tool when dealing with mathematical expressions involving powers. These rules simplify expressions and make them easier to work with. Here are some key exponent rules you should know:
  • Product of Powers Rule: When you multiply two powers with the same base, you add the exponents together. For example, \( a^m \times a^n = a^{m+n} \).
  • Power of a Power Rule: When raising an exponent to another power, you multiply the exponents. This means \( (x^m)^n = x^{m \cdot n} \).
  • Quotient Rule: This rule is used when dividing two powers with the same base. You subtract the exponent of the denominator from the exponent of the numerator: \( \frac{x^m}{x^n} = x^{m-n} \).
These rules form the foundation for simplifying algebraic expressions with exponents. By understanding and applying these rules, you can skillfully handle complex expressions.
Negative Exponents
Negative exponents can initially seem confusing, but they follow a simple principle. A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For example, \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \). This means that rather than multiplying a number by itself negatively, you divide 1 by the number raised to the positive version of the exponent.

Let's see how negative exponents play out in simplification:
  • When you encounter an expression such as \( a^{-3} \), this translates to \( \frac{1}{a^3} \).
  • If a negative exponent appears in a numerator or a denominator, use the reciprocal to make the exponent positive. For example, \( \frac{b^{-2}}{1} = \frac{1}{b^2} \).
Understanding negative exponents is essential for converting them into positive forms, as required by many algebraic expressions and problems.
Quotient Rule
The quotient rule is invaluable for simplifying expressions where you divide like bases with exponents. The formula \( \frac{x^m}{x^n} = x^{m-n} \) allows us to subtract the exponent of the denominator from the exponent of the numerator. This rule is straightforward: if the base is the same, you only need to focus on subtracting the exponents.

Here's an example of using the quotient rule:
  • Consider \( \frac{a^5}{a^3} \). Using the quotient rule, you have \( a^{5-3} = a^2 \).
  • If the exponent of the numerator is smaller, you end up with a negative exponent, such as \( \frac{a^2}{a^5} = a^{2-5} = a^{-3} \). This can further be rewritten as \( \frac{1}{a^3} \) using the rules for negative exponents.
The quotient rule simplifies complex fractions and makes it clear how the numerators and denominators interact when they share the same base. Utilize this rule to untangle various expressions efficiently.

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Most popular questions from this chapter

Perform the operations and simplify, if possible. See Example \(8 .\) $$\frac{x+5}{x y}-\frac{x-1}{x^{2} y}$$

Rain Gutters. A rectangular sheet of metal will be used to make a rain gutter by bending up its sides, as shown. If the ends are covered, the capacity \(f(x)\) of the gutter is a polynomial function of \(x: f(x)=-240 x^{2}+1,440 x .\) Find the capacity of the gutter if \(x\) is 3 inches. (GRAPH CANNOT COPY)

Perform the operations and simplify, if possible. See Example 9. $$\frac{3}{x+2}+\frac{5}{x-4}$$

In the following problems, simplify each expression by performing the indicated operations and solve each equation. $$\frac{34}{x^{2}}+\frac{13}{20 x}=\frac{3}{2 x}$$

Storage Tanks. The volume \(V(r)\) of the gasoline storage tank, in cubic feet, is given by the polynomial function \(V(r)=4.2 r^{3}+37.7 r^{2}\) where \(r\) is the radius in feet of the cylindrical part of the tank. What is the capacity of the tank if its radius is 4 feet?
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