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Chapter 1: Problem 32

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

Short Answer

Expert verified
\( t^{2 / 5} = \sqrt[5]{t^2} \)

Step by step solution

01

Identify the Exponent and Denominator

The given expression is in the form of an exponent. Identify the numerator and the denominator from the expression: \( t^{2 / 5} \). Here, 2 is the exponent, and 5 is the denominator.
02

Apply the Radical Notation Rule

Recall the rule that \( a^{m/n} = \sqrt[n]{a^m} \). This means that we can rewrite the expression in radical form using the denominator as the index of the root and the numerator as the power.
03

Convert to Radical Notation

Using the rule from Step 2, convert \( t^{2 / 5} \) to radical notation: \[ t^{2 / 5} = \sqrt[5]{t^2} \]

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Key Concepts

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

Exponent
Understanding exponents is essential when dealing with radical notation. An exponent is a number that indicates how many times a base is multiplied by itself. For example, in the expression \( t^{2} \), the base is 't', and the exponent is '2', meaning that 't' is multiplied by itself once (\( t \times t \)). Exponents can also be fractions, such as in the case of \( t^{2/5} \), where 2 is the numerator and 5 is the denominator. This fractional exponent can be interpreted in several ways, but one common method is through radical notation.
Denominator
In a fractional exponent, the denominator plays a crucial role. The denominator indicates the root of the base. For example, in the expression \( t^{2/5} \), the denominator is '5'. This means we need to take the fifth root of the base 't'. The conversion rule for fractional exponents to radical notation states that \( a^{m/n} \) can be written as \( \sqrt[n]{a^m} \). Thus, in our example, the fifth root of 't' is represented as \( \sqrt[5]{t} \). This denominator concept is essential for correctly converting between exponential and radical forms.
Root
The root in radical notation indicates which root of a number we are considering. The root can be identified from the denominator of the fractional exponent. For instance, in \( t^{2/5} \), the '5' in the denominator tells us to take the fifth root of 't'. Subsequently, the expression \( t^{2/5} \) converts into \( \sqrt[5]{t^2} \). Here, '2' is the exponent applied inside the radical, meaning we first raise 't' to the power of '2' and then take the fifth root of the result. Remember that:
  • The numerator (2) stays as the power
  • The denominator (5) becomes the root
. This transformation helps us to better understand and manipulate expressions involving fractional exponents.

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