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Chapter 8: Problem 82

Simplify. $$\frac{x^{\frac{7}{9}}}{x^{\frac{2}{9}}}$$

Short Answer

Expert verified
Simplified, it is \(x^{\frac{5}{9}}\).

Step by step solution

01

- Identify the Exponents

Recognize the exponents in the numerator and the denominator. In the given problem, the exponents are \(\frac{7}{9}\) and \(\frac{2}{9}\).
02

- Apply the Quotient Rule for Exponents

The quotient rule for exponents states that \(\frac{a^m}{a^n} = a^{m-n}\). Apply this rule to our exponents: \(\frac{x^{\frac{7}{9}}}{x^{\frac{2}{9}}} = x^{\frac{7}{9} - \frac{2}{9}}\).
03

- Subtract the Exponents

Subtract the exponents: \(\frac{7}{9} - \frac{2}{9} = \frac{5}{9}\).
04

- Simplify the Expression

Rewrite the expression using the simplified exponent: \(x^{\frac{5}{9}}\).

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Key Concepts

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

quotient rule
The Quotient Rule is essential for simplifying expressions involving exponents, especially when you have the same base. It states that \(\frac{a^m}{a^n} = a^{m-n}\). Here, \(a\) represents the base, while \(m\) and \(n\) are the exponents.
When using the Quotient Rule, you subtract the exponent in the denominator from the exponent in the numerator.
Let's walk through an example to make this clear. For the expression \(\frac{x^{\frac{7}{9}}}{x^{\frac{2}{9}}}\), we'll apply the Quotient Rule:
  • Identify the exponents: \(\frac{7}{9}\) and \(\frac{2}{9}\).
  • Subtract these exponents: \(\frac{7}{9} - \frac{2}{9} = \frac{5}{9}\).
  • Simplify the expression: \(x^{\frac{5}{9}}\).
This process streamlines simplifying complex algebraic expressions with exponents.
algebraic expressions
An algebraic expression contains numbers, variables, and arithmetic operations. Variables, typically represented by letters, can vary and take on different values. Understanding these components is crucial for manipulating expressions correctly.
In the expression \(\frac{x^{\frac{7}{9}}}{x^{\frac{2}{9}}}\), \(x\) is the variable. The exponents \(\frac{7}{9}\) and \(\frac{2}{9}\) are fractional. We dealt with this expression by applying the Quotient Rule.
To sum up:
  • Identify parts of the expression, such as variables and exponents.
  • Apply the appropriate rules, like the Quotient Rule, to simplify.
  • Carefully subtract the exponents and streamline your work.
These steps ensure you're working correctly through algebraic expressions and their transformations.
fractional exponents
Fractional exponents might look intimidating, but they follow the same rules as integer exponents.
A fractional exponent represents both a power and a root. For instance, \(x^{\frac{m}{n}}\) translates to the \(n\)th root of \(x\) raised to the \(m\)th power, or \(\sqrt[n]{x^m}\).
When simplifying \(\frac{x^{\frac{7}{9}}}{x^{\frac{2}{9}}}\), you treat the fractional exponents the same way as whole number exponents:
  • Apply the Quotient Rule: \(x^{\frac{7}{9}} - x^{\frac{2}{9}} = x^{\frac{5}{9}}\).
  • Realize \(x^{\frac{5}{9}}\) means \(\sqrt[9]{x^5}\).
Even though fractional exponents seem complex, they help express roots and powers concisely.
Remember, practice makes perfect, especially when dealing with these types of exponents.

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