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Chapter 8: Problem 81

Simplify. $$\frac{x^{\frac{4}{5}}}{x^{\frac{1}{5}}}$$

Short Answer

Expert verified
The simplified form is \[ x^{\frac{3}{5}} \].

Step by step solution

01

- Recall the property of exponents

When dividing like bases with exponents, subtract the exponents: \[ \frac{a^m}{a^n} = a^{m-n} \]
02

- Apply the property to the given expression

Using the property \[ \frac{x^{\frac{4}{5}}}{x^{\frac{1}{5}}} = x^{\frac{4}{5} - \frac{1}{5}} \]
03

- Simplify the exponent

Subtract the exponent in the denominator from the exponent in the numerator: \[ x^{\frac{4}{5} - \frac{1}{5}} = x^{\frac{3}{5}} \]

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Key Concepts

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

Exponent Rules
Exponents are a way to express repeated multiplication of a number by itself. The small number written as a superscript is called the exponent, and the large number is the base. There are several important rules when dealing with exponents that simplify mathematical expressions:
  • Product of Powers Rule: When multiplying two exponents with the same base, you add the exponents. For example, \(a^m * a^n = a^{m+n}\).
  • Power of a Power Rule: When raising an exponent to another exponent, you multiply the exponents. For instance, \( (a^m)^n = a^{m*n} \).
  • Quotient of Powers Rule: When dividing two exponents with the same base, you subtract the exponents, \( \frac{a^m}{a^n} = a^{m-n} \).
  • Zero Exponent Rule: Any non-zero number raised to the power of zero is 1, \( a^0 = 1 \).
  • Negative Exponent Rule: A negative exponent represents the reciprocal of the positive exponent, \( a^{-m} = \frac{1}{a^m} \).
Mastering these rules will help you manipulate and simplify expressions involving exponents with ease. Let鈥檚 apply the quotient of powers rule to our original problem.
Simplifying Expressions
Simplifying expressions involves rewriting them in a more concise, compact form without changing their value. Here are the steps to simplify our expression:
We start with:
\[ \frac{x^{\frac{4}{5}}}{x^{\frac{1}{5}}} \]
According to the quotient of powers rule, we subtract the exponent in the denominator from the exponent in the numerator:
\[ \frac{x^{\frac{4}{5}}}{x^{\frac{1}{5}}}= x^{\frac{4}{5}-\frac{1}{5}} \]
This results in:
\[ x^{\frac{3}{5}} \]
The simplified form of the expression is \( x^{\frac{3}{5}} \). By following these steps, you ensure that the expression is accurate and easy to understand.
Division of Powers
When dealing with the division of powers, it's essential to follow specific exponent rules which make it easier to handle. The primary rule in this context is the quotient of powers rule:
  • Quotient of Powers: This rule states that when you divide like bases, you subtract the exponent of the denominator from the exponent of the numerator, \( \frac{a^m}{a^n} = a^{m-n} \).
In our example:
\[ \frac{x^{\frac{4}{5}}}{x^{\frac{1}{5}}} \]
Applying the quotient of powers rule, it simplifies to:
\[ x^{\frac{4}{5} - \frac{1}{5}} \]
The exponent calculation would be:
\[ \frac{4}{5} - \frac{1}{5} = \frac{3}{5} \]
Thus, the simplified form is:
\[ x^{\frac{3}{5}} \]
Understanding and applying these rules helps solve complex problems involving exponents seamlessly.

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