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Chapter 0: Problem 105

In Exercises \(101-108,\) simplify by reducing the index of the radical. $$\sqrt[6]{x^{4}}$$

Short Answer

Expert verified
The simplified expression is \(\sqrt[3]{x^{2}}\).

Step by step solution

01

Apply the definition of the nth root

The nth root of a number can be written as that number raised to the power of 1 over n. Thus, the given expression \(\sqrt[6]{x^{4}}\) can be rewritten as \(x^{4/6}\).
02

Simplify the exponent

Next, simplify the fraction 4/6 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This will give \(x^{2/3}\).
03

Revert to the radical form

Finally, change the exponent back into the radical form. This results in the cube root of \(x^{2}\), which mathematically can be represented as \(\sqrt[3]{x^{2}}\).

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Key Concepts

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

Radical Notation
Radical notation is used to represent the 'nth root' of a number, which is the value that, when raised to the nth power, gives us the original number. It's written in the form \(\sqrt[n]{a}\), where \('a'\) is the radicand (the number under the radical sign) and \('n'\) is the index, indicating the root degree. For example, \(\sqrt[6]{x^{4}}\) indicates the 6th root of \('x^{4}'\).

Understanding radical notation is crucial in simplifying expressions and solving equations involving roots. We often work with radicals in algebra and advanced mathematics, making it essential for students to be comfortable with both the notation and the operations associated with roots.
Rational Exponents
Rational exponents provide an alternative way to express roots and powers. They follow the form \('a^{m/n}'\), where \('a'\) is the base, \('m'\) is the exponent, and \('n'\) is the root. The expression \('a^{m/n}'\) can be understood as \(\sqrt[n]{a^m}\), which is the nth root of \('a'\) raised to the mth power.

For instance, \(x^{4/6}\) is equivalent to \(\sqrt[6]{x^{4}}\) from our original exercise. Rational exponents allow us to simplify calculations and make it easier to perform operations like multiplication and division of exponents, which are common in algebra.
Nth Roots
When we talk about nth roots, we are referring to the inverse operation of raising a number to the power of 'n'. The nth root of a number \('x'\) is written as \(\sqrt[n]{x}\) and signifies a value that, when raised to the nth power, returns \('x'\). Common roots include the square root, where \('n=2'\), and the cube root, where \('n=3'\).

The exercise \(\sqrt[6]{x^{4}}\) is focused on finding the 6th root of \('x^{4}'\). Understanding how to manipulate and simplify nth roots is very important in higher-level math, including algebra, calculus, and beyond.
Index Reduction
Index reduction is a simplification process involving radicals, where the goal is to reduce the index of the radical without changing the value of the expression. This process is often achieved by converting the radical into a rational exponent, simplifying the exponent if possible, and then potentially converting back into a radical with a smaller index.

In our example, the 6th root of \('x^{4}'\) was simplified to \(\sqrt[3]{x^{2}}\) by reducing the index from 6 to 3. This allowed for a simpler expression and made it easier to work with the radical. Students should learn to recognize opportunities for index reduction to simplify complex radical expressions.

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Most popular questions from this chapter

Rationalize the numerator. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$\begin{aligned} &2>1\\\ &2(y-x)>1(y-x)\\\ &2 y-2 x>y-x\\\ &\begin{aligned} y-2 x &>-x \\ y &>x \end{aligned} \end{aligned}$$ The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Rationalize the numerator. $$\frac{\sqrt{x}-\sqrt{y}}{x^{2}-y^{2}}$$

Exercises \(142-144\) will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$ 2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right) $$
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