Chapter 13: Problem 42
Challenge Problems. $$\left(\frac{5 a^{5} b^{3}}{2 x^{2} y}\right)^{3 n}$$
Short Answer
Expert verified
\[\left(\frac{5^{3n} a^{15n} b^{9n}}{2^{3n} x^{6n} y^{3n}}\right)\]
Step by step solution
01
Simplify the Expression
To simplify the expression \(\left(\frac{5 a^{5} b^{3}}{2 x^{2} y}\right)^{3 n}\), apply the power rule of exponents, which states that \(\left(\frac{a}{b}\right)^{n} = \frac{a^n}{b^n}\).
02
Raise Numerator and Denominator to Power
Raise both the numerator \(5 a^{5} b^{3}\) and the denominator \(2 x^{2} y\) to the power of \(3n\), separately: \((5 a^{5} b^{3})^{3n} \text{ and } (2 x^{2} y)^{3n}\).
03
Apply the Power of a Product Rule
Use the power of a product rule, which states that \(\left(ab\right)^n = a^n b^n\), to each term in the numerator and denominator separately to get \(5^{3n} (a^5)^{3n} (b^3)^{3n} \text{ in the numerator and } 2^{3n} (x^2)^{3n} y^{3n} \text{ in the denominator}\).
04
Simplify Individual Exponents
Further simplify by using \(\left(a^m\right)^n = a^{mn}\) to combine the exponents: \( 5^{3n} a^{15n} b^{9n} \text{ for the numerator and } 2^{3n} x^{6n} y^{3n} \text{ for the denominator}\).
05
Combine the Result
Combine the simplified numerator and denominator to rewrite the expression: \[\left(\frac{5^{3n} a^{15n} b^{9n}}{2^{3n} x^{6n} y^{3n}}\right)\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplifying Algebraic Expressions
Understanding how to simplify algebraic expressions is essential for students studying algebra. The process involves reducing expressions to their simplest form, making them easier to work with. Simplification might include combining like terms, using the distributive property, or applying the power rules of exponents.
For instance, let's consider the expression \(\left(\frac{5 a^{5} b^{3}}{2 x^{2} y}\right)^{3 n}\). The goal is not only to make the expression more elegant but also to prepare it for further manipulation or solving. To begin simplifying, we first identify that within the parentheses we have a fraction raised to an exponent of \(3n\). By understanding the properties of exponents, we can approach the simplification step by step, applying specific rules like the power rule of exponents, which allows us to raise separate terms to a power independently.
Further tips on simplification involve writing intermediate steps and keeping track of signs and coefficients. This prevents errors and increases clarity. Also, students are encouraged to familiarize themselves with exponent laws fully to ensure they apply them correctly.
For instance, let's consider the expression \(\left(\frac{5 a^{5} b^{3}}{2 x^{2} y}\right)^{3 n}\). The goal is not only to make the expression more elegant but also to prepare it for further manipulation or solving. To begin simplifying, we first identify that within the parentheses we have a fraction raised to an exponent of \(3n\). By understanding the properties of exponents, we can approach the simplification step by step, applying specific rules like the power rule of exponents, which allows us to raise separate terms to a power independently.
Further tips on simplification involve writing intermediate steps and keeping track of signs and coefficients. This prevents errors and increases clarity. Also, students are encouraged to familiarize themselves with exponent laws fully to ensure they apply them correctly.
Power of a Product Rule
The power of a product rule is an imperative concept when dealing with exponentiation in algebra. This rule tells us that when a product of bases is raised to an exponent, the exponent applies to each base separately. Mathematically, this is expressed as \(\left(ab\right)^n = a^n b^n\).
Applying this rule becomes catalytic when simplifying the complex expression \(\left(\frac{5 a^{5} b^{3}}{2 x^{2} y}\right)^{3 n}\). We separate the components of the numerator and denominator respectively and raise each to the power of \(3n\). This precise step is crucial because it reduces the complicated expression into more manageable parts, leading to \(5^{3n} (a^5)^{3n} (b^3)^{3n}\) for the numerator and \(2^{3n} (x^2)^{3n} y^{3n}\) for the denominator.
To apply this rule correctly, it's helpful to remember that multiplication before raising to a power is a must. Forget this, and you run the risk of incorrect simplification. Practice with various expressions can help ingrained this rule deeply.
Applying this rule becomes catalytic when simplifying the complex expression \(\left(\frac{5 a^{5} b^{3}}{2 x^{2} y}\right)^{3 n}\). We separate the components of the numerator and denominator respectively and raise each to the power of \(3n\). This precise step is crucial because it reduces the complicated expression into more manageable parts, leading to \(5^{3n} (a^5)^{3n} (b^3)^{3n}\) for the numerator and \(2^{3n} (x^2)^{3n} y^{3n}\) for the denominator.
To apply this rule correctly, it's helpful to remember that multiplication before raising to a power is a must. Forget this, and you run the risk of incorrect simplification. Practice with various expressions can help ingrained this rule deeply.
Exponential Expressions
Exponential expressions, such as \(a^{n}\), where \(a\) is the base and \(n\) the exponent, represent repeated multiplication. In the given exercise, we encounter more complex exponential expressions that include several bases and variable exponents.
The ultimate simplification of \(\left(\frac{5 a^{5} b^{3}}{2 x^{2} y}\right)^{3 n}\) leads to individual exponents for each base: \(5^{3n} a^{15n} b^{9n}\) over \(2^{3n} x^{6n} y^{3n}\). This is the result of understanding and applying another exponent rule, \(\left(a^m\right)^n = a^{mn}\), which indicates that a power to a power can be simplified by multiplying the exponents.
Practice is crucial when dealing with exponential expressions. Understanding how to manipulate exponents not only simplifies expressions but also prepares students for solving equations and understanding functions involving growth and decay. Always ensure that the base and its respective exponent are clear to avoid misinterpretation of the expression's value.
The ultimate simplification of \(\left(\frac{5 a^{5} b^{3}}{2 x^{2} y}\right)^{3 n}\) leads to individual exponents for each base: \(5^{3n} a^{15n} b^{9n}\) over \(2^{3n} x^{6n} y^{3n}\). This is the result of understanding and applying another exponent rule, \(\left(a^m\right)^n = a^{mn}\), which indicates that a power to a power can be simplified by multiplying the exponents.
Practice is crucial when dealing with exponential expressions. Understanding how to manipulate exponents not only simplifies expressions but also prepares students for solving equations and understanding functions involving growth and decay. Always ensure that the base and its respective exponent are clear to avoid misinterpretation of the expression's value.
