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Chapter 4: Problem 25

If \(x>0, \frac{\sqrt[3]{x^{2}}}{\sqrt[6]{x}}=?\) (A) \(\sqrt[3]{x} \quad\) OR (B) \(\sqrt{x}\)

Short Answer

Expert verified
Option (B): \( \sqrt{x} \).

Step by step solution

01

Express Using Exponents

First, express the expression using exponents. We have \( \sqrt[3]{x^2} = x^{2/3} \) and \( \sqrt[6]{x} = x^{1/6} \). The original expression becomes \( \frac{x^{2/3}}{x^{1/6}} \).
02

Apply the Quotient Rule for Exponents

Use the quotient rule for exponents which states \( \frac{a^m}{a^n} = a^{m-n} \). Our expression \( \frac{x^{2/3}}{x^{1/6}} \) can be simplified to \( x^{2/3 - 1/6} \).
03

Common Denominator

Find a common denominator to subtract the exponents: \( \frac{2}{3} - \frac{1}{6} = \frac{4}{6} - \frac{1}{6} = \frac{3}{6} \).
04

Simplify the Exponent

Simplify \( \frac{3}{6} \) to \( \frac{1}{2} \). Therefore, the expression becomes \( x^{1/2} \), which is \( \sqrt{x} \).
05

Identify the Correct Option

Compare the result \( \sqrt{x} \) with the given options. This matches option (B).

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Key Concepts

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

Exponents
Exponents are a powerful mathematical tool that represent the process of raising a base number to a specific power. This power indicates how many times the base is multiplied by itself. For example, the expression \( a^b \) means that the base \( a \) is being multiplied by itself \( b \) times. Exponents are fundamental in simplifying expressions, especially when dealing with large numbers or complex algebraic expressions. They allow us to condense repeated multiplication into a more manageable form.

Some basic properties of exponents include:
  • \( a^m \times a^n = a^{m+n} \) (Product Rule)
  • \( \left(a^m\right)^n = a^{m \times n} \) (Power of a Power Rule)
  • \( a^m / a^n = a^{m-n} \) (Quotient Rule, which we will discuss further)
  • \( a^0 = 1 \) for any non-zero \( a \)
Understanding these rules is crucial as they provide a strong foundation for tackling various mathematical problems involving exponents.
Quotient Rule for Exponents
The quotient rule for exponents is a greatly useful tool when simplifying expressions involving division of powers with the same base. According to this rule, if you have the expression \( \frac{a^m}{a^n} \), it simplifies to \( a^{m-n} \). This rule helps to eliminate the base from the denominator by subtracting the exponent in the denominator from the exponent in the numerator.

To apply the quotient rule effectively:
  • Identify that both the numerator and the denominator share the same base.
  • Subtract the exponent in the denominator from the exponent in the numerator to find the exponent of the base in the resulting expression.
For example, with the expression \( \frac{x^{2/3}}{x^{1/6}} \), you would use the quotient rule to simplify it to \( x^{2/3 - 1/6} = x^{1/2} \). This simplification makes it easier to evaluate or compare with other expressions.
Simplifying Radicals
Simplifying radicals involves expressing a radical expression in its simplest form. Radicals often look complicated, but with the right techniques, they can be simplified to expressions involving exponents for easier manipulation. A radical expression such as \( \sqrt[n]{x^m} \) can be converted to an expression using exponents like \( x^{m/n} \).

When simplifying radicals:
  • Convert the radical to an exponent form using the rule \( \sqrt[n]{x^m} = x^{m/n} \). This makes it easier to apply rules of exponents.
  • Once in exponent form, simplify the expression using exponent rules such as the quotient rule, product rule, or power of a power rule.
  • Finally, convert back to radical form if needed, or keep it in a simplified exponent form depending on the requirement of the problem.
Applying these steps to the original problem, \( \sqrt[3]{x^{2}} \) can be rewritten as \( x^{2/3} \) and \( \sqrt[6]{x} \) as \( x^{1/6} \). This transformation is key to simplifying the expression further using the quotient rule.

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