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Chapter 7: Problem 2

Write each expression in radical form. $$ x^{\frac{2}{3}} $$

Short Answer

Expert verified
The expression \( x^{\frac{2}{3}} \) is \( \sqrt[3]{x^2} \) in radical form.

Step by step solution

01

Understand Exponents and Radical Notation

The expression given is \( x^{\frac{2}{3}} \). This expression indicates a fractional exponent, which can be converted into radical form. A fractional exponent such as \( a^{\frac{m}{n}} \) is equivalent to the \( n\)-th root of \( a \) raised to the \( m\)-th power.
02

Rewrite Using Radical Notation

To convert \( x^{\frac{2}{3}} \) into radical form, identify that \( 3 \) is the root and \( 2 \) is the power. Hence, if \( x^{\frac{m}{n}} = \sqrt[n]{x^m} \), then \( x^{\frac{2}{3}} = \sqrt[3]{x^2} \). This means take the cube root of \( x \) and then square the result.
03

Confirm the Radical Form

In radical form, \( x^{\frac{2}{3}} \) is written as \( \sqrt[3]{x^2} \), confirming the conversion process from exponent notation to radical notation. This represents the same mathematical operation as the original expression.

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Key Concepts

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

Fractional Exponents
Fractional exponents are a way of expressing powers and roots together in a concise format. When you see an expression like \( a^{\frac{m}{n}} \), it signifies a two-step operation: taking the \( n\)-th root of \( a \) and then raising the result to the \( m\)-th power.

This notation helps in simplifying complex expressions and finding simpler ways to handle roots and powers that appear frequently in algebra. Here are some key points:
  • The numerator (\( m \)) of a fractional exponent corresponds to the power.
  • The denominator (\( n \)) signifies the root.
For example, \( x^{\frac{2}{3}} \) represents the cube root of \( x \) squared. This transformation into fractional exponents aids better manipulation of algebraic expressions.
Radical Notation
Radical notation is the way in which roots are expressed, traditionally with the radical sign (\( \sqrt{} \)). Understanding radical notation involves recognizing how different parts of a radical expression relate. When rewriting an expression like \( x^{\frac{2}{3}} \) in radical form, you use both the root indicated by the denominator and the power from the numerator.

The process might look like this:
  • The number underneath the radical sign is the base (here, \( x \)).
  • The index of the radical (here, \( 3 \)) represents the root.
  • Exponentiation of the result completes the operation (raising to the power of \( 2 \)).
Thus, \( x^{\frac{2}{3}} \) becomes \( \sqrt[3]{x^2} \), showing that you first find the cube root of \( x \) and then square the outcome.
Exponent Conversion
Converting between exponential and radical forms is a valuable skill in algebra. This conversion involves recognition of how fractional exponents translate into radical expressions. With this understanding, you can change any expression from one form to another seamlessly.
  • A general expression \( a^{\frac{m}{n}} \) converts to \( \sqrt[n]{a^m} \).
  • Recognize that swapping forms does not alter the value of the expression, just the way it's expressed.
For instance, in the expression \( x^{\frac{2}{3}} \), recognizing the fractional exponent's components allows you to convert it into a radical representation: \( \sqrt[3]{x^2} \). Mastery of exponent conversion helps simplify the process of solving equations and manipulating expressions involving roots and powers effectively.

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Most popular questions from this chapter

Solve each inequality. $$ 1+\sqrt{7 x-3} > 3 $$

Find \((f+g)(x),(f-g)(x),(f \cdot g)(x),\) and \(\left(\frac{f}{g}\right)(x)\) for each \(f(x)\) and \(g(x)\) $$ \begin{array}{l}{f(x)=x+5} \\ {g(x)=x-3}\end{array} $$

Solve each inequality. $$ \sqrt{c+5}+\sqrt{c+10} > 2 $$

REVIEW Which of the following sentences is true about the graphs of \(y=2(x-3)^{2}+1\) and \(y=2(x+3)^{2}+1 ?\) F Their vertices are maximums G The graphs have the same shape with different vertices. H The graphs have different shapes with different vertices. J One graph has a vertex that is a maximum while the other graph has a vertex that is a minimum.

Solve each equation. $$ \sqrt{x}=4 $$
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