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Chapter 5: Problem 116

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

Short Answer

Expert verified
\( \frac{n^{3}}{125} \) is the simplified expression.

Step by step solution

01

Understand the Expression

The expression given is \( \left( \frac{n}{5} \right)^{3} \). This is a fraction raised to the power of 3.
02

Apply the Power of a Quotient Rule

According to the power of a quotient rule, \( \left( \frac{a}{b} \right)^{n} = \frac{a^{n}}{b^{n}} \). Therefore, apply this rule to the expression: \( \left( \frac{n}{5} \right)^{3} = \frac{n^{3}}{5^{3}} \).
03

Simplify the Denominator

The denominator is \(5^{3}\). Calculate the cube of 5: \(5^{3} = 5 \times 5 \times 5 = 125\).
04

Write the Simplified Expression

Combine the results from the previous steps: \( \frac{n^{3}}{125} \). This is the simplified form of the given expression.

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Key Concepts

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

Understanding the Power of a Quotient Rule
When dealing with expressions like \( \left( \frac{a}{b} \right)^{n} \), you are tasked with applying the "power of a quotient rule". This rule is essential for simplifying complex algebraic fractions raised to a power. It states that when a fraction is raised to a power, both the numerator and the denominator must be raised to that same power. For example, for the expression \( \left( \frac{n}{5} \right)^{3} \), apply the power to both \(n\) and \(5\) separately as follows:
  • Numerator: \( n^{3} \)
  • Denominator: \( 5^{3} \)
This yields the new fraction \( \frac{n^{3}}{5^{3}} \). This formula is a crucial tool that saves time and reduces error by directly providing a way to handle fractional exponents.
Techniques for Simplifying Expressions
Simplifying expressions is a fundamental part of algebra that transforms complex equations into more manageable forms. Simplification involves:
  • Applying mathematical rules or laws, such as the power of a quotient rule.
  • Performing arithmetic operations to compute values, like determining \(5^{3}\).
  • Using common sense to combine like terms and reduce fractions.
These steps are quintessential when working on polynomial expressions or any algebraic forms. For instance, in the expression \( \frac{n^{3}}{125} \) derived from \( \left( \frac{n}{5} \right)^{3} \), you simplify by first applying the aforementioned rule and then calculating \(5^{3} = 125\), resulting in a neat fraction that is easy to interpret.
Exploring Fraction Exponents
Fraction exponents, also known as rational exponents, can seem tricky but they offer a powerful shorthand in mathematics. They tell you how to handle roots and powers simultaneously. For example, the expression \( a^{\frac{m}{n}} \) can mean several things:
  • \( \sqrt[n]{a^{m}} \) - take the nth root of \(a^{m}\).
  • \( (\sqrt[n]{a})^{m} \) - first find the nth root of \(a\) and then raise it to the power of \(m\).
Fraction exponents let you write complex root and power combinations more succinctly. This way, rather than working through multiple steps manually, you can use indices to get the answers quickly. When combined with rules like the power of a quotient, they enable fluency in manipulating and solving algebraic equations effectively.

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Most popular questions from this chapter

a. Find each of the following products. i. \((x-1)(x+1)\) ii. \((x-1)\left(x^{2}+x+1\right)\) iii. \((x-1)\left(x^{3}+x^{2}+x+1\right)\) b. Write a product of two polynomials such that the result is \(x^{5}-1\)

Perform the operations. $$ (4 b-8)^{2} $$

Perform the operations. $$ (3 m-2 n)^{3} $$

a. Find two binomials whose product is a binomial. b. Find two binomials whose product is a trinomial. c. Find two binomials whose product is a four-term polynomial.

Perform each division. $$ \frac{8 x^{9}-32 x^{6}}{4 x^{4}} $$
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