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Chapter 1: Problem 19

Convert to expressions with rational exponents. \(\sqrt[5]{a^{3}}\)

Short Answer

Expert verified
The rational exponent form is \(a^{3/5}\).

Step by step solution

01

Understand the radical expression

The given expression is a radical expression where we have a fifth root of a cubed variable, \(\root{5}{a^{3}}\). Our goal is to convert this into an expression with rational exponents.
02

Recall the conversion rule for radicals

The rule for converting a radical to a rational exponent is \(\root{n}{x} = x^{1/n}\). For a radical like \(\root{n}{x^m}\), it converts to \(x^{m/n}\).
03

Apply the conversion rule

In our expression, \(\root{5}{a^{3}}\), we will use the rule from Step 2. Here, \(n = 5\) and \(m = 3\). Thus, \(\root{5}{a^{3}} = a^{3/5}\).

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Key Concepts

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

headline of the respective core concept
Radical expressions can often look complicated, but understanding their structure makes them more approachable. A radical expression includes a root symbol with a number under it, like \(\root{n}{x}\). For example, the exercise uses \(\root{5}{a^{3}}\), which means the fifth root of \(a^{3}\). When dealing with radical expressions, it's crucial to get comfortable with the notation and what it signifies in relation to exponents.
**Common Components:**
  • The number under the root symbol is the radicand (here, it is \(a^{3}\)).
  • The small number outside the root symbol (the index) tells you which root you're dealing with (in this case, 5).
As you familiarize yourself with these terms and their roles, converting radical expressions into more manageable forms becomes easier. This leads to the next important concept: **Conversion rules**.
headline of the respective core concept
To simplify radical expressions, converting them to rational exponents is a powerful tool. The conversion rule states that a radical \( \root{n}{x}\) can be written as \( x^{1/n}\). When dealing with a radical that includes an exponent, like \(\root{n}{x^m}\), it translates to \(\root{n}{x^m}= x^{m/n}\).
**Example from Exercise:**
  • Given: \(\root{5}{a^{3}}\)
  • Conversion: The index is 5 and the exponent is 3, so the expression becomes \(\root{5}{a^{3}} = a^{3/5}\).
By applying these conversion rules, it is easier to manipulate and solve problems involving radical expressions.
headline of the respective core concept
Exponent laws are essential for working with expressions and simplifying calculations. When converting radicals to rational exponents, certain rules govern how these exponents behave.
**Key Exponent Laws:**
  • **Multiplication:** \(a^m \times a^n = a^{m+n}\) - When multiplying like bases, you add their exponents.
  • **Division:** \(a^m / a^n = a^{m-n}\) - When dividing like bases, you subtract the exponents.
  • **Power of a Power:** \( (a^m)^n = a^{mn}\) - When taking a power to another power, you multiply the exponents.
These laws are the backbone of simplifying and solving expressions involving exponents. Knowing and utilizing them will make handling rational exponents straightforward and less intimidating. Applying these in different combinations allows for flexibility and simplification in complex problems.

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Most popular questions from this chapter

Find all solutions of the given equation. $$ \cos ^{2} x+5 \cos x=-6 $$

Show that \(1+\tan ^{2} t=\sec ^{2} t\). (Hint: Begin with the Pythagorean Identity of Theorem 6 and divide both sides by \(\cos ^{2} t\).)

Use a calculator to find the values of the following trigonometric functions. \(\cot (3 \pi / 11)\)

a) Show that \(\cos 2 t=\cos ^{2} t-\sin ^{2} t\) b) Show that \(\cos 2 t=2 \cos ^{2} t-1\). (Hint: Use part (a) and a Pythagorean Identity.)

Sketch the following angles. \(2 \pi\)
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