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Chapter 5: Problem 105

Simplify each expression, if possible. $$ \left(\frac{3 m^{4}}{2 n^{5}}\right)^{5} $$

Short Answer

Expert verified
The simplified expression is \(\frac{243m^{20}}{32n^{25}}\).

Step by step solution

01

Apply the Power of a Quotient Rule

According to the power of a quotient rule, \[\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n},\]we distribute the exponent 5 to both the numerator and the denominator. This gives us:\[\left( \frac{3m^4}{2n^5} \right)^5 = \frac{(3m^4)^5}{(2n^5)^5}.\]
02

Apply the Power of a Product Rule

For the numerator \((3m^4)^5\), use the power of a product rule, \((ab)^n = a^n b^n\):\[(3m^4)^5 = 3^5 \cdot (m^4)^5.\]Similarly, for the denominator \((2n^5)^5\), \[(2n^5)^5 = 2^5 \cdot (n^5)^5.\] So this simplifies our expression to:\[\frac{3^5 \cdot (m^4)^5}{2^5 \cdot (n^5)^5}.\]
03

Calculate the Powers

Now, calculate all the powers separately:- For the constants: - Calculate \(3^5 = 243\) - Calculate \(2^5 = 32\)- For the variables: - Use the power of a power rule, \((a^m)^n = a^{m \cdot n}\), to find: - \((m^4)^5 = m^{4\cdot 5} = m^{20}\) - \((n^5)^5 = n^{5\cdot 5} = n^{25}\) After calculating, we have:\[\frac{243 \cdot m^{20}}{32 \cdot n^{25}}.\]
04

Simplified Expression

The expression is now simplified:\[\frac{243m^{20}}{32n^{25}}.\]There are no common factors in the numerator and denominator, so this is the simplest form.

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Key Concepts

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

Power of a Quotient Rule
When simplifying algebraic expressions, the power of a quotient rule is a valuable tool. This rule states: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). It means that when you raise a fraction to a power, you apply the exponent to both the numerator and the denominator separately.
Here's how you use it:
  • Take a fraction inside the parentheses, such as \( \left( \frac{3m^4}{2n^5} \right) \).
  • Apply the exponent, which in this case is 5, to both the numerator (\(3m^4\)) and the denominator (\(2n^5\)).
  • The result is \( \frac{(3m^4)^5}{(2n^5)^5} \).
This simplifies the process by dealing with each part of the fraction separately, making it easier to handle larger or more complex expressions. Remember this rule whenever you need to raise a fraction to a power.
Power of a Product Rule
The power of a product rule is another handy rule for simplifying expressions. This states: \((ab)^n = a^n b^n\). It tells us that when you need to raise a product to a power, you apply the exponent to each factor within the product separately.
Let's break it down:
  • Consider the expression \( (3m^4)^5 \).
  • Using the power of a product rule, separate the numbers and variables: raise 3 and \(m^4\) individually to the power of 5.
  • This results in \(3^5 \cdot (m^4)^5\).
Similarly, for the denominator \((2n^5)^5\), we split it into individual components: \(2^5 \cdot (n^5)^5\). This approach simplifies our expression into manageable parts, making it easier to compute each part and eventually combine them back together.
Power of a Power Rule
The power of a power rule helps when simplifying expressions with nested exponents. It states: \((a^m)^n = a^{m \cdot n}\). When you have a power raised to another power, you multiply the exponents.
Here's how to use this rule:
  • For an expression like \((m^4)^5\), apply this rule by multiplying the exponents: \(m^{4 \cdot 5} = m^{20}\).
  • Similarly, for \((n^5)^5\), multiply the exponents to get \(n^{5 \cdot 5} = n^{25}\).
This rule simplifies computations involving powers, turning complex exponents into straightforward multiplication. When used together with the other rules, it simplifies the entire expression to create a tidy final result.

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