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Chapter 1: Problem 34

Convert to radical notation. \(y^{-2 / 3}\)

Short Answer

Expert verified
\( \frac{1}{\sqrt[3]{y^2}} \)

Step by step solution

01

Identify the terms

The given expression is in the form of an exponent. Here, the base is y, the numerator of the exponent is -2, and the denominator of the exponent is 3.
02

Understand the significance of the negative exponent

A negative exponent indicates that the base should be taken as a reciprocal. So, we can rewrite the expression as \(\frac{1}{y^{2/3}}\).
03

Convert fractional exponent to radical form

A fractional exponent represents a root. Specifically, the denominator of the fractional exponent (3) represents the root, while the numerator (2) represents the power. Therefore, \(y^{2/3}\) can be written as \(\sqrt[3]{y^2}\).
04

Combine the reciprocal and radical form

Putting together the reciprocal and the radical form, the expression becomes \(\frac{1}{\sqrt[3]{y^2}}\).

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Key Concepts

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

Negative Exponents
A negative exponent flips the base to its reciprocal. For example, if you have an expression like \(a^{-n}\), you can rewrite it as \(\frac{1}{a^n}\). This principle is crucial in simplifying expressions with negative exponents.

Key points about negative exponents:
  • A negative exponent indicates division, not multiplication.
  • The base of the exponent stays the same; only its position in the fraction changes.
  • It helps simplify expressions and solve complex algebraic problems.
Fractional Exponents
Fractional exponents might seem tricky, but they are pretty straightforward once broken down. They represent both powers and roots. For instance, in \(a^{m/n}\), the numerator (m) is the power, and the denominator (n) is the root.

Major points on fractional exponents:
  • The denominator tells you the root, while the numerator tells you the power.
  • \(a^{m/n}\) can be rewritten as \(\sqrt[n]{a^m}\).
  • Understanding fractional exponents makes it easier to convert between exponential and radical forms.
Radical Expressions
Radical expressions involve roots, like square roots or cube roots. Converting between radical and exponential forms is often needed to simplify these expressions.

Important aspects of radical expressions:
  • The radical sign (鈭�) indicates the root.
  • For cube roots, we use \(\sqrt[3]{a}\).
  • \(\sqrt[n]{a^m}\) is the same as \(a^{m/n}\).
  • This relationship helps in solving algebraic equations involving roots.

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Most popular questions from this chapter

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