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Chapter 6: Problem 395

Simplify. $$\left(\frac{3 m^{5}}{5 n}\right)^{3}$$

Short Answer

Expert verified
\( \frac{27 m^{15}}{125 n^3} \)

Step by step solution

01

Apply the Power Rule to Each Component

Raise both the numerator and the denominator to the third power. Use the formula \[(\frac{a}{b})^n = \frac{a^n}{b^n}\] to get \[\left(\frac{3 m^5}{5 n}\right)^3 = \frac{(3 m^5)^3}{(5 n)^3}.\]
02

Simplify the Numerator

Raise each part of the numerator \[(3 m^5)\] to the third power separately: \[3^3 = 27 \] and \[ (m^5)^3 = m^{15}.\] This gives us \[ (3 m^5)^3 = 27 m^{15}. \]
03

Simplify the Denominator

Raise each part of the denominator \[(5 n)\] to the third power separately: \[5^3 = 125\] and \[n^3 = n^3.\] This gives us \[(5 n)^3 = 125 n^3. \]
04

Combine the Simplified Components

Combine the simplified numerator and denominator: \[ \frac{27 m^{15}}{125 n^3} \]

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Key Concepts

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

Power Rule
The power rule is essential to understand when simplifying expressions like \((\frac{3 m^5}{5 n})^3\). It tells us how to apply an exponent to every part of an expression raised to a power. Follow these steps:
  • Raise each component in the numerator to the given power.
  • Do the same for each component in the denominator.
Do you remember the formula \((\frac{a}{b})^n = \frac{a^n}{b^n}\)? This formula helps simplify expressions: raise the numerator and the denominator to the power separately. This means: \((\frac{3 m^5}{5 n})^3\) becomes \((\frac{3^3 \cdot (m^5)^3}{5^3 \cdot n^3})\). Using the power rule correctly will make the process of simplification much clearer and organized.
Exponents
Exponents are a shorthand way of expressing repeated multiplication. For example, \[3^3 = 3 \cdot 3 \cdot 3 = 27.\] This concept is used to simplify components individually within an algebraic expression. In the given exercise, we raise each part of \[3m^5\] and \[5n\] to the power of three:
  • \(3^3 = 27\)
  • \((m^5)^3 = m^{15}\)
When multiplying powers of the same base, you add the exponents. This is why \[ (m^5)^3 \] becomes \[ m^{15} \.\] Understanding exponents and their rules helps in performing such multiplications with ease and accuracy.
Rational Expressions
Rational expressions are fractions that have algebraic expressions in their numerator and denominator. In our exercise, \((\frac{3 m^5}{5 n})^3\) is a rational expression raised to a power. Simplifying rational expressions involves several steps:
  • Applying the power rule.
  • Simplifying the numerator separately.
  • Simplifying the denominator separately.
It's important to work systematically. Simplify each part of the fraction before combining them: raise the \[3m^5\] and \[5n\] separately to the power of three, and then divide the resulting terms. Finally, recombine the simplified numerator and denominator. By following these steps, the expression simplifies to \((\frac{27 m^{15}}{125 n^3}).\)
Numerator and Denominator
Understanding numerators and denominators is crucial when working with fractions and rational expressions. The numerator is the top part of a fraction, and the denominator is the bottom part. In our given exercise,
  • The numerator is \[(3 m^5)\]
  • The denominator is \[(5 n)\]
When we apply the power rule, we need to handle each separately: The numerator \[ (3 m^5)^3 \] turns into \[(27 m^{15})\], and the denominator \[ (5 n)^3 \] becomes \[ (125 n^3).\] By simplifying part by part, especially within complex expressions, we maintain clarity and accuracy. Finally, combining the results yields the simplified expression \[(\frac{27 m^{15}}{125 n^3}).\]

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Most popular questions from this chapter

Divide each polynomial by the monomial. $$\frac{4 w^{2}+2 w-5}{2 w}$$

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