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Chapter 8: Problem 102

Write each expression in radical form. $$ t^{\frac{2}{3}} $$

Short Answer

Expert verified
\(\sqrt[3]{t^2}\)

Step by step solution

01

Identify the Root

First, identify the denominator of the fractional exponent. In \(t^{2/3}\), the denominator is 3. This tells us that we will be using a cube root in the conversion to radical form.
02

Identify the Power

Next, identify the numerator of the fractional exponent. In \(t^{2/3}\), the numerator is 2. This tells us that in the radical form, t will be squared (鈭�2).
03

Write in Radical Form

Lastly, rewrite the expression in radical form. The cube root is represented by the symbol 鈭�, and the power is represented by squaring (or putting a 2 exponent). Therefore, \(t^{2/3}\) in radical form equals \(\sqrt[3]{t^2}\).

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Key Concepts

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

Fractional Exponents
Fractional exponents are an intriguing way to express roots and powers within the same notation. These exponents are fractions, where the numerator stands for the power and the denominator denotes the root.
In the expression \(t^{\frac{2}{3}}\), \(\frac{2}{3}\) is the fractional exponent.
It's key to know that:
  • The numerator (2) indicates that \(t\) is multiplied by itself, or squared.
  • The denominator (3) specifies a cube root.
Thus, fractional exponents provide a dual function of root and power, simplifying many mathematical operations. Instead of separate root and power operations, everything combines into a single operation for ease of computation.
Cube Root
The cube root is an essential concept when dealing with radical expressions, especially when the fraction's denominator in an exponent is 3. The cube root of a number \(x\) is the number that, when multiplied by itself three times, gives \(x\).
For instance, to write \(t^{\frac{2}{3}}\) in radical form, the denominator 3 suggests we use the cube root, represented as \(\sqrt[3]{ \cdot }\).
So, the expression becomes \(\sqrt[3]{t^2}\).
  • Cube roots are useful for simplifying higher degree polynomials.
  • They help in expressing fractional powers as radicals.
Understanding cube roots allows you to handle expressions with fractional exponents effortlessly.
Exponentiation
Exponentiation involves raising a number or variable to a power, indicated by the exponent. This process, when applied, signifies multiplying the base by itself as many times as the exponent specifies. In \(t^{\frac{2}{3}}\), exponentiation affects the base \(t\).
Here, the number 2 signifies squaring the variable \(t\).
Combining exponentiation with roots, as is done in radicals, creates expressions like \(\sqrt[3]{t^2}\).
  • Exponentiation simplifies repeated multiplication.
  • It aids in solving equations involving powers and roots.
  • Combining roots and powers offers a comprehensive tool for mathematical computation.
Through understanding the principles of exponentiation, math becomes less daunting and more manageable, especially when tackling complex fractional exponents.

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Most popular questions from this chapter

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