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Chapter 10: Problem 24

Use radical notation to write each expression. Simplify if possible. $$ 2 x^{3 / 5} $$

Short Answer

Expert verified
The expression in radical form is \(2 \cdot \sqrt[5]{x^3}\).

Step by step solution

01

Understand the Expression

The expression given is \(2x^{3/5}\). We need to rewrite this expression using radical notation. In a general sense, an expression like \(x^{m/n}\) is equivalent to \(\sqrt[n]{x^m}\).
02

Apply Radical Notation

Using the formula \(x^{m/n} = \sqrt[n]{x^m}\), we can rewrite \(x^{3/5}\) as \(\sqrt[5]{x^3}\). Therefore, the expression \(2x^{3/5}\) can be rewritten as \(2 \cdot \sqrt[5]{x^3}\).
03

Simplify Further if Possible

The expression \(2 \cdot \sqrt[5]{x^3}\) is already simplified using radical notation. There is no further simplification possible since there are no like terms or factors that can be simplified further.

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Key Concepts

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

Exponents
Exponents are a crucial part of algebra and are used to represent repeated multiplication. If you have an expression like \( x^a \), it means that \( x \) is multiplied by itself \( a \) times. In our case, with \( x^{3/5} \), the exponent is in fractional form, which implies a combination of power and root. Exponents can help simplify expressions and make calculations easier, especially when dealing with large numbers or complex equations.
  • The base tells us what number is being multiplied.
  • The exponent tells us how many times to multiply the base by itself.
Understanding how to manipulate exponents, including converting between fractional and radical notation, is key in algebraic simplification and solving equations.
Simplification
Simplification in mathematics involves rewriting an expression to make it easier to read or work with, without changing its value. For the expression \( 2x^{3/5} \), simplification can involve rewriting it in a form that is more understandable, like using radical notation. The idea is to present the expression in the simplest form that maintains the same value. When simplifying expressions:
  • Look for like terms that can be combined.
  • Apply the rules of exponents to combine or break down terms.
  • Convert complex expressions to simpler ones using algebraic identities and notation. For example, \( x^{3/5} \) is simplified to \( \sqrt[5]{x^3} \), making it a radical expression.
Effective simplification helps in solving equations quickly and accurately, especially when dealing with large expressions.
Fractional Exponents
Understanding fractional exponents is essential because they bridge the connection between powers and roots. A fractional exponent like \( x^{m/n} \) means the nth root of \( x^m \). It鈥檚 another way to express radicals. In our exercise, \( x^{3/5} \) translates to \( \sqrt[5]{x^3} \), indicating the fifth root of \( x^3 \). Here鈥檚 what fractional exponents signify:
  • The numerator \( m \) represents the power to which the base is raised before taking the root.
  • The denominator \( n \) indicates the root to be used.
  • This allows for complex operations through simple rules of exponents, even involving noninteger powers.
Mastering fractional exponents can greatly enhance your skills in algebra, offering a powerful tool for solving equations involving roots and powers.

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Most popular questions from this chapter

Which of the following are not real numbers? $$\sqrt[10]{-17}$$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[6]{4} $$

In electronics, the angular frequency of oscillations in a certain type of circuit is given by the expression \((L C)^{-1 / 2}\) Use radical notation to write this expression.

Which of the following are not real numbers? Explain why \(\sqrt{-64}\) is not a real number.

Use a calculator to write a four-decimal-place approximation of each number. $$ 76^{5 / 7} $$
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