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Chapter 4: Problem 3

Simplify where possible. Express your answer with positive exponents. a. \(\frac{2^{3} x^{4}}{2^{5} x^{8}}\) b. \(\frac{x^{4} y^{7}}{x^{3} y^{-5}}\) c. \(\frac{x^{-2} y}{x y^{3}}\) d. \(\frac{(x+y)^{4}}{(x+y)^{-7}}\) e. \(\frac{a^{-2} b c^{-5}}{\left(a b^{2}\right)^{-3} c}\)

Short Answer

Expert verified
(a) \( \frac{1}{4 x^{4}} \); (b) \( x y^{12} \); (c) \( \frac{1}{x^{3} y^{2}} \); (d) \( (x+y)^{11} \); (e) \( \frac{a b^{7}}{c^{6}} \)

Step by step solution

01

Part a: Simplify the Fraction

Simplify \( \frac{2^{3} x^{4}}{2^{5} x^{8}} \).Use the quotient rule for exponents: \( \frac{a^m}{a^n} = a^{m-n} \).So, \( \frac{2^{3}}{2^{5}} = 2^{3-5} = 2^{-2} \) and \( \frac{x^{4}}{x^{8}} = x^{4-8} = x^{-4} \).Combine these results: \( 2^{-2} x^{-4} \).Express with positive exponents: \( \frac{1}{2^{2}} \frac{1}{x^{4}} = \frac{1}{4x^{4}} \).
02

Part b: Simplify the Fraction

Simplify \( \frac{x^{4} y^{7}}{x^{3} y^{-5}} \).Use the quotient rule for exponents: \( \frac{a^m}{a^n} = a^{m-n} \).So, \( \frac{x^{4}}{x^{3}} = x^{4-3} = x^{1} = x \) and \( \frac{y^{7}}{y^{-5}} = y^{7-(-5)} = y^{7+5} = y^{12} \).Combine these results: \( x y^{12} \).
03

Part c: Simplify the Fraction

Simplify \( \frac{x^{-2} y}{x y^{3}} \).Use the quotient rule for exponents: \( \frac{a^m}{a^n} = a^{m-n} \).So, \( \frac{x^{-2}}{x} = x^{-2-1} = x^{-3} \) and \( \frac{y}{y^{3}} = y^{1-3} = y^{-2} \).Combine these results: \( x^{-3} y^{-2} \).Express with positive exponents: \( \frac{1}{x^{3}} \frac{1}{y^{2}} = \frac{1}{x^{3} y^{2}} \).
04

Part d: Simplify the Fraction

Simplify \( \frac{(x+y)^{4}}{(x+y)^{-7}} \).Use the quotient rule for exponents: \( \frac{a^m}{a^n} = a^{m-n} \).So, \( \frac{(x+y)^4}{(x+y)^{-7}} = (x+y)^{4-(-7)} = (x+y)^{4+7} = (x+y)^{11} \).
05

Part e: Simplify the Fraction

Simplify \( \frac{a^{-2} b c^{-5}}{(a b^{2})^{-3} c} \).First, simplify the denominator: \( (a b^{2})^{-3} = a^{-3} (b^{2})^{-3} = a^{-3} b^{-6} \).So, the original expression becomes: \( \frac{a^{-2} b c^{-5}}{a^{-3} b^{-6} c} \).Use the quotient rule for exponents for each variable:For \( a \): \( \frac{a^{-2}}{a^{-3}} = a^{-2-(-3)} = a^{-2+3} = a^{1} = a \).For \( b \): \( \frac{b}{b^{-6}} = b^{1-(-6)} = b^{1+6} = b^{7} \).For \( c \): \( \frac{c^{-5}}{c} = c^{-5-1} = c^{-6} \). Express \( c^{-6} \) with a positive exponent: \( \frac{1}{c^6} \).So, the expression becomes: \( a b^{7} \frac{1}{c^{6}} = \frac{a b^{7}}{c^{6}} \).

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Key Concepts

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

Quotient Rule for Exponents
The quotient rule for exponents helps you simplify expressions involving division of similar bases. This is expressed as: \(\frac{a^m}{a^n} = a^{m-n}\). Essentially, when dividing two exponents with the same base, you subtract the power in the denominator from the power in the numerator.
For example, simplifying \(\frac{2^3}{2^5}\) follows as: \(2^{3-5} = 2^{-2}\). This subtraction method greatly simplifies complex algebraic expressions.
Understanding this rule is critical as it forms the foundation for dealing with fractions involving exponents in algebra.
Positive Exponents
In algebra, it鈥檚 generally cleaner and more intuitive to express answers with positive exponents. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For instance, \(2^{-2}\) equals \(\frac{1}{2^2} = \frac{1}{4}\).
Whenever you encounter a negative exponent, convert it to a positive exponent by moving the base to the opposite part of the fraction. For example, \(x^{-3} = \frac{1}{x^3}\). Expressing in positive exponents makes it easier to interpret and work with the expressions.
Simplify Fractions
When simplifying fractions in algebra, combine the rules of exponents with traditional fraction simplification techniques. Simplifying involves reducing the fractions by canceling common terms and applying exponent rules.
Take \(\frac{x^{-2} y}{x y^3}\) as an example. Apply the quotient rule first: \(\frac{x^{-2}}{x} = x^{-3}\) and \(\frac{y}{y^3} = y^{-2}\). This simplifies the fraction to \(\frac{1}{x^3 y^2}\). Give each component separately and then combine for a fully simplified fraction, ensuring positive exponents wherever possible.
Algebraic Expressions
Algebraic expressions consist of variables, constants, and operations combined together. Simplifying these expressions often involves applying multiple rules and techniques.
Consider the expression \(\frac{a^{-2} b c^{-5}}{(a b^2)^{-3} c}\). First, simplify the components in the denominator: \( (a b^2)^{-3} = a^{-3} b^{-6} \). Then, apply the quotient rule for each variable: \( \frac{a^{-2}}{a^{-3}} = a\), \(\frac{b}{b^{-6}} = b^7\), and \(\frac{c^{-5}}{c} = c^{-6}\). The result, \(\frac{a b^7}{c^6}\), is a simplified algebraic expression.
Mastering these foundational concepts can greatly improve your proficiency in algebra.

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Most popular questions from this chapter

Use \(<,>,\) or \(=\) to make each statement true. a. \(\sqrt{3}+\sqrt{7} \underline{?} \sqrt{3+7}\) b. \(\sqrt{3^{2}+2^{2}} \underline{?} 5\) c. \(\sqrt{5^{2}-4^{2}} \geq 2\)

Round off the numbers and then estimate the values of the following expressions without a calculator. Show your work, writing your answers in scientific notation. If available, use a calculator to verify your answers. a. (0.000359)\(\cdot(0.00000247)\) b. \(\frac{0.00000731 \cdot 82,560}{1,891,000}\)

An angstrom, \(\AA,\) is a metric unit of length equal to one ten billionth of a meter. It is useful in specifying wavelengths of electromagnetic radiation (e.g., visible light, ultraviolet light, X-rays, and gamma rays). a. The visible-light spectrum extends from approximately 3900 angstroms (violet light) to 7700 angstroms (red light) Write this range in centimeters using scientific notation. b. Some gamma rays have wavelengths of 0.0001 angstrom. Write this number in centimeters using scientific notation. c. The nanometer (nm) is 10 times larger than the angstrom, so \(1 \mathrm{nm}\) is equal to how many meters?

Compare the times listed below by plotting them on the same order-of-magnitude scale. (Hint: Start by converting all the times to seconds.) a. The time of one heartbeat ( 1 second) b. Time to walk from one class to another ( 10 minutes) c. Time to drive across the country ( 7 days) d. One year (365 days) e. Time for light to travel to the center of the Milky Way \((38,000\) years \()\) f. Time for light to travel to Andromeda, the nearest large galaxy ( 2.2 million years)

Without using a calculator, find two consecutive integers such that one is smaller and one is larger than each of the following (for example, \(3<\sqrt{11}<4\) ). Show your reasoning. a. \(\sqrt{13}\) b. \(\sqrt{22}\) c. \(\sqrt{40}\)
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