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Chapter 1: Problem 33

Convert to radical notation. . \(t^{-2 / 5}\)

Short Answer

Expert verified
\(t^{-2/5} = \frac{1}{\root{5}{t^2}}\).

Step by step solution

01

Understand Exponent Notation

Recognize that the given expression, \(t^{-2/5}\), is in exponent notation. The base of the expression is \(t\) and the exponent is \(-2/5\).
02

Apply the Negative Exponent Rule

Recall the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^n}\). Apply this rule to the expression to get: \(t^{-2/5} = \frac{1}{t^{2/5}}\).
03

Convert to Radical Notation

Recall that an exponent in the form of \(a^{m/n}\) can be written in radical notation as \(\root{n}{a^m}\). Apply this rule to \(t^{2/5}\): \(t^{2/5} = \root{5}{t^2}\).
04

Combine and Simplify

Combine the results to rewrite the expression in radical notation. Thus, \(\frac{1}{t^{2/5}} = \frac{1}{\root{5}{t^2}}\).

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Key Concepts

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

Negative Exponent Rule
Negative exponents can be a bit tricky at first, but they're quite easy once you understand the rule. If you have an expression like \(a^{-n}\), where \(a\) is any number (or variable) and \(n\) is a positive integer, you can rewrite it using the negative exponent rule. This rule states that \(a^{-n} = \frac{1}{a^n}\). So, you're basically flipping the fraction. Let's break it down:
  • Start with \(t^{-2/5}\).
  • The \(-2/5\) exponent means we need to take the reciprocal to make the exponent positive.
  • Applying the rule, we convert \(t^{-2/5}\) to \(\frac{1}{t^{2/5}} \).
This step is crucial because it prepares us to handle the rational exponent in a simpler form. Always remember, a negative exponent just flips the base into the denominator.
Rational Exponents
Rational exponents might look complicated, but they simply represent roots. A rational exponent is one that is a fraction, like \(2/5\). The general form \(a^{m/n}\) can be interpreted in radical notation as \(\root{n}{a^m}\). Here鈥檚 how it works:
  • If \(t^{2/5}\) confuses you, break it down.
  • The denominator (5) represents the root you'll take, in this case, the fifth root.
  • The numerator (2) tells you to square what's inside the root.
So, \(t^{2/5} = \root{5}{t^2}\). Now you know that a rational fraction acts like a combination of roots and powers. It's just another way to write radical expressions. This makes it easier to manage complex problems by breaking them into more digestible parts.
Radical Expressions
Radical expressions involve roots, such as square roots or cube roots. Converting from a rational exponent to radical notation is straightforward once you understand the components. Here's how we deal with our expression \(t^{-2/5} \:=> \frac{1}{t^{2/5}}\):
  • We already know \(t^{2/5} \) is \(\root{5}{t^2}\).
  • Combine the rules: \( \frac{1}{t^{2/5}} \) becomes \( \frac{1}{\root{5}{t^2}}\).
So, we go from a rational exponent to a more familiar radical form. This method makes it easier to visualize and simplify complex expressions. Understanding radical notation is vital for higher-level maths, as it frequently appears in calculus and algebra.

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Most popular questions from this chapter

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