Question

Rewrite the expression \(a^{-s / t}\) in radical form. Then state the index of the radical.

Short answer

The expression \(a^{-s / t}\) can be rewritten in radical form as \( \frac{1}{\sqrt[t]{a^s}}\) . The index of the radical is \(t\).

Step-by-step solution

Understand the Parts of the Fractional Exponent

An expression of the form \(a^{m/n}\) can be rewritten in radical form as the n-th root of \(a^m\). Additionally, if the exponent is negative, the expression will be the reciprocal of the base raised to the positive exponent.

Apply the Rules of Exponents to Convert into Radical Form

The negative sign indicates that the expression is the reciprocal. So, \(a^{-s/t}= \frac{1}{a^{s/t}}\). Then, rewrite the exponent as a root: \(a^{s/t}= \sqrt[t]{a^s}\). Therefore, the original expression can be written as \( \frac{1}{\sqrt[t]{a^s}}\).

Identify the Index of the Radical

The index of the radical is the number that signifies the degree of the root, which in this case is \(t\).

Key concepts

Radical Form

When you encounter a term with a fractional exponent, like \(a^{m/n}\), rewriting it in radical form is a handy technique to understand the expression better. The radical form translates the expression into a root, making it more intuitive.

• The denominator of the fractional exponent becomes the index of the radical or the degree of the root.
• The numerator, on the other hand, remains as the power to which the base is raised.

So, if you have \(a^{s/t}\), it converts to \(\sqrt[t]{a^s}\). This format is sometimes easier to handle because roots are familiar mathematical concepts. Furthermore, understanding the radical form can help simplify expressions or solve equations where square roots or cube roots naturally occur.

Negative Exponents

Negative exponents might initially seem tricky, but they follow a straightforward rule. A negative exponent inverts the base. So, \(a^{-n}\) flips the base into its reciprocal form, turning into \(\frac{1}{a^n}\).This concept holds true for fractional exponents as well. For instance, \(a^{-s/t}\) can be rewritten by first handling the negative exponent. This means you take the reciprocal of the base raised to the absolute value of the exponent: \(\frac{1}{a^{s/t}}\).Once the expression is in reciprocal form, the next step is converting into radical form. Knowing how to deal with negative exponents is essential because they often appear in simplifying complex algebraic expressions or solving equations.

Index of the Radical

The index of a radical is an important part of understanding radical expressions. Essentially, the index is the number that indicates which root you are taking.

• If the index is 2, you are dealing with a square root, typically denoted as \(\sqrt{a}\).
• An index of 3 implies a cube root, as seen in \(\sqrt[3]{a}\).
• For any positive integer \(t\), \(\sqrt[t]{a}\) indicates the n-th root of \(a\).

In expressions like \(a^{s/t}\), the denominator \(t\) of the fractional exponent directly translates into the index of the radical. It tells you the degree of the root to be calculated, such as whether it's a square root, cube root, or any nth root. Always identifying the index first can make it easier to rewrite expressions in radical form and solve related problems.