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Chapter 0: Problem 105

Simplify by reducing the index of the radical. $$\sqrt[6]{x^{4}}$$

Short Answer

Expert verified
The simplified form of \( \sqrt[6]{x^{4}} \) is \( x^{\frac{2}{3}} \)

Step by step solution

01

Identify the Index and the exponent

Here, the index of the given radical is 6 while the exponent of x is 4. Thus, n=6 and m=4.
02

Apply the Rule of Indices

The basic rule of indices states \( \sqrt[n]{x^{m}} = x^{\frac{m}{n}} \). If we substitute n=6 and m=4 into this rule, we will get \( \sqrt[6]{x^{4}} = x^{\frac{4}{6}} \).
03

Simplify the Exponent

The fraction \( \frac{4}{6} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (gcd), which is 2. Doing this gives \( \frac{4}{6} = \frac{2}{3} \). Thus, \( x^{\frac{4}{6}} = x^{\frac{2}{3}} \).

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Key Concepts

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

Index of Radical
The index of a radical tells us the degree of the root we are dealing with. Radicals are expressions that involve roots, such as square roots, cube roots, etc.
In the expression \( \sqrt[n]{x^m} \), the index is represented by \( n \). This indicates that the root we are taking is the \( n \)-th root.
  • For instance, in \( \sqrt{a} \), we are dealing with a square root, where the index is implicitly 2.
  • In \( \sqrt[3]{b} \), the index is 3, indicating a cube root.
  • In our original exercise \( \sqrt[6]{x^{4}} \), the index is 6, meaning we are looking for the sixth root of \( x^{4} \).
Knowing the index is crucial because it determines how the expression can be simplified or recalculated in terms of exponents. Identifying the index correctly is the first step when you are dealing with radical simplification.
Rule of Indices
The rule of indices offers a powerful framework for simplifying expressions involving exponents. It establishes that both indices and radicals can change format while retaining their mathematical integrity.
According to the rule, a radical such as \( \sqrt[n]{x^m} \) can be rewritten as an exponent: \( x^{\frac{m}{n}} \).
This conversion process is essential for simplifying expressions, as it allows us to leverage our understanding of exponents to do so.
  • For example, converting \( \sqrt[6]{x^4} \) to its exponential form results in \( x^{\frac{4}{6}} \), a form where simplification becomes more intuitive.
  • This method capitalizes on the fraction \( \frac{m}{n} \), which represents the power \( m \) being divided across \( n \).
Applying this rule bridges the gap between radical notation and fractional exponents, facilitating the process of simplification.
Simplifying Exponents
Once we've represented our radical expression with fractional exponents, the next step is to simplify those exponents for a more straightforward result. The process entails reducing the fractional exponent to its simplest form.
Simplifying involves finding the greatest common divisor (gcd) of the numerator and the denominator and dividing both by it.
  • In \( x^{\frac{4}{6}} \), the gcd of 4 and 6 is 2.
  • Dividing both top and bottom of the fraction by 2, we get \( x^{\frac{2}{3}} \).
This simplification takes advantage of pure arithmetic to reduce the expression to its simplest possible form.
It's a fundamental step not only for clarity but also for further computational ease in algebraic manipulations.

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Most popular questions from this chapter

In Exercises \(1-16,\) evaluate each algebraic expression for the given value or values of the variable(s). $$\frac{2 x+3 y}{x+1}, \text { for } x=-2 \text { and } y=4$$

In Exercises \(121-128,\) write each English phrase as an algebraic expression. Then simplify the expression. Let \(x\) represent the number. A number decreased by the difference between eight and the number

Simplify each complex rational expression. $$\frac{\frac{3}{x-2}-\frac{4}{x+2}}{\frac{7}{x^{2}-4}}$$

What is a polynomial in \(x ?\)

Write each number in decimal notation without the use of exponents. $$-7.16 \times 10^{6}$$
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