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Chapter 8: Problem 123

Write with a rational exponent. (a) \(\sqrt[7]{x}\) (b) \(\sqrt[9]{y} (c) \sqrt[5]{f}\)

Short Answer

Expert verified
(a) x^{1/7} (b) y^{1/9} (c) f^{1/5}

Step by step solution

01

Understand radical expressions

Radical expressions such as \(\textbackslash sqrt[n]{a}\) can be rewritten using rational exponents. The general form for converting a radical expression to an exponent is \(a^{1/n}\), where \(n\) is the index of the radical.
02

Convert \(\textbackslash sqrt[7]{x}\) to rational exponent

Using the rule mentioned, \(\textbackslash sqrt[7]{x}\) can be rewritten as \(\textbackslash x^{1/7}\).
03

Convert \(\textbackslash sqrt[9]{y}\) to rational exponent

Similarly, \(\textbackslash sqrt[9]{y}\) can be rewritten as \(\textbackslash y^{1/9}\).
04

Convert \(\textbackslash sqrt[5]{f}\) to rational exponent

Following the same rule, \(\textbackslash sqrt[5]{f}\) can be rewritten as \(\textbackslash f^{1/5}\).

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Key Concepts

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

Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, and so on. The symbol used is the radical sign (鈭�), and the number under the radical sign is called the radicand. For example, in \( \textbackslash sqrt[7]{x} \), 'x' is the radicand. The expression can also have a 'degree' or 'index', which tells you which root you're taking. If the index is not specified, it is assumed to be 2 (a square root). Understanding how to work with radical expressions is important for simplifying algebraic equations and expressions.
Rational Exponents
Rational exponents offer an alternative way to represent roots. Instead of using a radical sign, a rational exponent is used in the format \(a^{1/n}\), where 'a' is the base and 'n' is the index of the root. This conversion makes it easier to use the properties of exponents to simplify or manipulate expressions. For instance, \( \textbackslash sqrt[7]{x} \) can be written as \( x^{1/7} \). Similarly, \( \textbackslash sqrt[9]{y} \) and \( \textbackslash sqrt[5]{f} \) can be written as \( y^{1/9} \) and \( f^{1/5} \), respectively.
  • Converting radicals to rational exponents makes calculations simpler
  • Use properties of exponents for easier manipulation
Index of Radical
The index of a radical indicates the degree of the root. For example, in \( \textbackslash sqrt[3]{8} \), the '3' is the index, specifying that it is a cube root. If no index is written, it is assumed to be 2 (a square root). The index is crucial when converting radicals to rational exponents.
  • An index of 'n' in \( \textbackslash sqrt[n]{a} \) means the expression is equivalent to \( a^{1/n} \)
  • Helps in simplifying expressions and solving equations involving roots

Converting the index into a fraction helps leverage the power rules of exponents, making complex algebraic problems easier to tackle.

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Most popular questions from this chapter

Explain what is meant by the \(n^{t h}\) root of a number.

Simplify. Assume all variables are positive (a) \(\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}\) (b) \(\left(a^{\frac{1}{3}} b^{\frac{2}{3}}\right)^{\frac{3}{2}}\)

Simplify. (a) \(8 \sqrt{3 c}+2 \sqrt{3 c}-9 \sqrt{3 c}\)(b)\(2 \sqrt[3]{4 p q}-5 \sqrt[3]{4 p q}+4 \sqrt[3]{4 p q}\)

Simplify using absolute value signs as needed. (a) \(\sqrt[5]{-32}\) (b) \(\sqrt[8]{-1}\)

Simplify. Assume all variables are positive (a) \(\frac{c^{\frac{5}{3}} \cdot c^{-\frac{1}{3}}}{c^{-\frac{2}{3}}}$$\left(\frac{8 x^{\frac{5}{3}} y^{-\frac{1}{2}}}{27 x^{-\frac{4}{3}} y^{\frac{5}{2}}}\right)^{\frac{1}{3}}\)
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