Problem 15 - Algebra 2

Question

Write each expression in radical form. $$ t^{-\frac{3}{4}} $$

Step-by-Step Solution

Answer

The expression in radical form is \(\frac{1}{\sqrt[4]{t^3}}\).

Step 1: Identify negative exponent

Looking at the expression \(t^{-\frac{3}{4}}\), the negative exponent indicates a reciprocal. Reciprocal means that \(t^{-\frac{3}{4}} = \frac{1}{t^{\frac{3}{4}}}\)

Step 2: Identify fractional exponent

In the expression \(\frac{1}{t^{\frac{3}{4}}}\), the fractional exponent can be broken down where the numerator represents the power and the denominator represents the root. Therefore, the fractional expression can be written as \(\frac{1}{(t^3)^{\frac{1}{4}}}\)

Step 3: Rewrite in radical form

We can now rewrite the fractional expression in radical form. Therefore, we have our final answer: \(\frac{1}{\sqrt[4]{t^3}}\), which represents the fourth root of \(t^3\).

Key Concepts

Negative Exponents

Dealing with negative exponents might seem tricky at first, but once you get the hang of it, it becomes simple. A negative exponent shows that you need to find the reciprocal of the base raised to the corresponding positive exponent.
For example, consider the expression \( t^{-\frac{3}{4}} \). The negative sign tells us that we need to take the reciprocal.
This means you can change it from \( t^{-\frac{3}{4}} \) to \( \frac{1}{t^{\frac{3}{4}}} \).

• The key takeaway is: switch the position of the base from numerator to denominator or vice versa and make the exponent positive.
• Negative exponents do not make a number negative: they merely "flip" where the base resides in terms of a fraction.

Fractional Exponents

Fractional exponents represent both roots and powers within a single expression. The numerator indicates the power, while the denominator shows the root.
In our expression \( \frac{1}{t^{\frac{3}{4}}} \), \(\frac{3}{4}\) can be viewed as taking the cube of \(t\), and then the fourth root of that result.

• The structure is:
  • Numerator (\(3\)): This is the power, meaning you raise the base \(t\) to the power of \(3\).
  • Denominator (\(4\)): This corresponds to the root, indicating the fourth root.

Understanding this, our expression \( t^{\frac{3}{4}} \) becomes \((t^3)^{\frac{1}{4}}\). This breakdown makes it easier to transform the expression into radical form.

Reciprocal

The concept of a reciprocal ties closely with negative exponents. When converting an expression like \( t^{-\frac{3}{4}} \) into a form without a negative exponent, we use its reciprocal: \( \frac{1}{t^{\frac{3}{4}}} \).

• The reciprocal of a number or expression \( a \) is \( \frac{1}{a} \).
• It "flips" the position of the base, changing the base from the numerator to the denominator.
  • For example, the reciprocal of \(2\) is \(\frac{1}{2}\).
• This step does not alter the value of the original expression but changes how it is represented.

In the context of negative exponents, using the reciprocal helps eliminate the negative sign, simplifying further operations on the expression.

Radical Form Conversion

Converting expressions from a fractional exponent form to a radical form involves understanding the representation of roots. A fractional exponent like \( \frac{3}{4} \) has already been broken down to \((t^3)^{\frac{1}{4}}\), implying the fourth root of \(t^3\).

• The term "radical" involves a root, represented by a radical sign \(\sqrt{}\).
• The expression \((t^3)^{\frac{1}{4}}\) can be written in radical form as \(\sqrt[4]{t^3}\).
• Radical forms help
  • Visualize the operation (root) more clearly.
  • Provide an easily interpretable format, especially useful for solving further equations.

Thus, the original expression \( t^{-\frac{3}{4}} \) in radical form becomes \( \frac{1}{\sqrt[4]{t^3}} \), which explicitly states the operation as the reciprocal of the fourth root of \(t^3\).