Problem 13 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The radical form of the expression \( y^{\frac{2}{3}} \) is \( \sqrt[3]{y^2} \).

Step 1: Identify the Exponent and Root From the Given Expression

In the expression \( y^{\frac{2}{3}} \), the numerator of the fraction represents the exponent (power) which is 2, while the denominator represents the root which is 3.

Step 2: Convert to Radical Form

Using the general rule for converting from exponential to radical form, we replace the base and its fractional exponent with a radical. Therefore, the expression \( y^{\frac{2}{3}} \) converts to \( \sqrt[3]{y^2} \), which is the radical form.

Key Concepts

Exponents

Exponents are a powerful mathematical tool that helps us express large numbers or repeated multiplication in a compact form. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \( y^2 \), \( y \) is the base, and 2 is the exponent, telling us to multiply \( y \) by itself: \( y \times y \).
Exponents simplify reading and writing calculations involving repeated multiplication.

• If \( y = 3 \), then \( y^2 = 3 \times 3 = 9 \).
• The expression gives us a quick way to calculate powers without repetitive tasks.

Understanding exponents is crucial for working with more complex expressions, such as those involving fractional exponents.

Fractional Exponents

Fractional exponents combine the ideas of roots and powers into a single operation. Instead of using the traditional square root or cube root notation, fractional exponents express these operations more efficiently. For instance, \( y^{\frac{1}{2}} \) is the same as the square root of \( y \), or \( \sqrt{y} \).
In a fractional exponent \( b^{\frac{m}{n}} \):

• The numerator \( m \) is the power applied to the base.
• The denominator \( n \) indicates the root we are taking.

Returning to our example, \( y^{\frac{2}{3}} \) tells us to first take the cube root of \( y \) and then square the result:

• Cube root: \( \sqrt[3]{y} \)
• Square the result: \( (\sqrt[3]{y})^2 = \sqrt[3]{y^2} \)

This dual operation can be a bit confusing initially, but practicing with examples solidifies the concept.

Radical Form

When converting from fractional exponents to radical form, it's crucial to understand how radicals represent roots undertaken by a number. The radical symbol \( \sqrt{} \) signifies the action of taking a root, where the index of the radical indicates which root to take. If no index is specified, it is typically understood to be a square root.
In our example, converting \( y^{\frac{2}{3}} \) involves expressing it in radical form as \( \sqrt[3]{y^2} \). This means:

• The index \( 3 \) shows it is a cube root, as specified by the denominator of the fractional exponent.
• The power of \( y^2 \) is applied under the radical.

Using radical form provides a visual representation of the operation being performed, illustrating both the power and the root. It helps make complex expressions more comprehensible and manageable, especially when performing algebraic operations.