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Chapter 6: Problem 6

\(y=\frac{1}{\sqrt[3]{x}}\)

Short Answer

Expert verified
The function \(y = \frac{1}{\sqrt[3]{x}}\) can be simplified to \(y = x^{-\frac{1}{3}}\).

Step by step solution

01

- Identify the Function

Look at the given function: \(y = \frac{1}{\sqrt[3]{x}}\). This is a function that has a radical in the denominator with a cube root.
02

- Rewrite the Denominator

Rewrite the cube root in the denominator using a rational exponent. The cube root of \(x\) can be written as \(x^{\frac{1}{3}}\), so the function becomes \(y = \frac{1}{x^{\frac{1}{3}}}\).
03

- Apply the Negative Exponent Rule

Use the rule of exponents that \(a^{-n} = \frac{1}{a^n}\). Applying this rule here, \(y = x^{-\frac{1}{3}}\).

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Key Concepts

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

Exponent Rules
When dealing with calculus problems, understanding exponent rules is essential. These rules help simplify expressions and make computations easier. Here are a few crucial exponent rules to remember:
  • Multiplying Powers with the Same Base: \[a^m \times a^n = a^{m+n}\]
  • Dividing Powers with the Same Base: \[\frac{a^m}{a^n} = a^{m-n}\]
  • Power of a Power: \[(a^m)^n = a^{m \times n}\]
  • Negative Exponent Rule: \[a^{-n} = \frac{1}{a^n}\]
  • Zero Exponent Rule: \[a^0 = 1\]

Applying these rules can transform complex radical expressions into more manageable terms. For instance, in our exercise, we applied the negative exponent rule to convert \[\frac{1}{x^{\frac{1}{3}}}\] into \[x^{-\frac{1}{3}}\]. This transformation simplifies further calculus operations like differentiation or integration.
Rational Exponents
Rational exponents can sometimes be confusing because they combine properties of both exponents and roots. A rational exponent like \[x^{\frac{m}{n}}\] involves both taking a root and raising to a power:
  • The denominator () indicates the root. For example, \[x^{\frac{1}{3}}\] means the cube root of \x\.
  • The numerator () indicates the power. For instance, \[x^{\frac{2}{3}}\] means first taking the cube root of \x\, then squaring the result.

In the given problem, the cube root of \x\ can be written as \[x^{\frac{1}{3}}\]. This shows how rational exponents provide an alternative notation for radicals, which is especially useful in calculus. Simplifying radicals to rational exponents can make differentiation and integration processes more straightforward.
Functions with Radicals
Functions with radicals, such as the square root or cube root, often appear in calculus. These functions can initially seem complex, but re-writing them with rational exponents is a helpful strategy. For example:
\[\frac{1}{\text{cube root of } x} = \frac{1}{x^{1/3}}\]
By understanding how to convert between radicals and rational exponents, calculus problems involving roots become much easier to handle. This allows you to apply various differentiation and integration techniques more effectively.
When tackling these functions, always:
  • Simplify the radicals using rational exponents.
  • Apply exponent rules to further simplify the expression.
  • Remember that these transformations make it easier to use fundamental calculus operations.

Mastering these techniques will enhance your ability to solve complex calculus problems efficiently.

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Most popular questions from this chapter

A contractor agrees to paint on both sides of 1000 circular signs each of radius \(3 \mathrm{ft}\). Upon receiving the signs, it is discovered that the radius is \(\frac{1}{2}\) in. too large. Use differentials to find the approximate percent increase of paint that will be needed.

At any point \((x, y)\) on a curve, \(D_{x}^{3} y=2\), and \((1,3)\) is a point of inflection at which the slope of the inflectional tangent is \(-2 .\) Find an equation of the curve.

\(\int\left(4 x^{3}-3 x^{2}+6 x-1\right) d x\)

\(\frac{1}{\sqrt[4]{15}}\)

\(3 x^{3}-x^{2} y+2 x y^{2}-y^{3}-3 x^{2}+y^{2}=1\)
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