Problem 7 - Intermediate Algebra

Question

Complete the table by writing the given expression in the alternate form. Also give the base and exponent for the exponential form. \begin{tabular}{|c|c|c|c|} \hline Radical form & Exponential form & Base & Exponent \\ \hline\(\sqrt[5]{25}\) & & & \\ \((\sqrt[4]{16})^{-3}\) & \((-27)^{2 / 3}\) & & \\ & & \\ \(-\sqrt{\frac{9}{64}}\) & & & \\ \hline \end{tabular}

Step-by-Step Solution

Answer

Converted radicals to exponents: (\sqrt[5]{25}) = 25^{1/5}; (\sqrt[4]{16})^{-3} = 16^{-3/4}; -\sqrt{\frac{9}{64}} = -\left(\frac{9}{64}\right)^{1/2}.

Step 1: Convert Radical Form to Exponential Form

To convert a radical expression to an exponential form, we use the property \( \sqrt[n]{a} = a^{1/n} \). For \( \sqrt[5]{25} \), the exponential form is \( 25^{1/5} \).

Step 2: Identify Base and Exponent - First Expression

For the expression \( 25^{1/5} \), the base is 25 and the exponent is \( \frac{1}{5} \).

Step 3: Convert Nested Radical and Power

The expression \( (\sqrt[4]{16})^{-3} \) can be converted in steps: first, \( \sqrt[4]{16} \) becomes \( 16^{1/4} \). Raise the exponent to the power of -3: \( (16^{1/4})^{-3} \), which simplifies to \( 16^{-3/4} \).

Step 4: Identify Base and Exponent - Second Expression

For the expression \( 16^{-3/4} \), the base is 16 and the exponent is \( \frac{-3}{4} \).

Step 5: Identify Base and Exponent - Third Expression

For the already provided expression \( (-27)^{2/3} \), the base is -27 and the exponent is \( \frac{2}{3} \).

Step 6: Convert Square Root Form

The expression \( -\sqrt{\frac{9}{64}} \) is the same as \( -\left(\frac{9}{64}\right)^{1/2} \).

Step 7: Identify Base and Exponent - Fourth Expression

For the expression \( -\left(\frac{9}{64}\right)^{1/2} \), the base is \( \frac{9}{64} \) and the exponent is \( \frac{1}{2} \).

Step 8: Filling in the Table

Based on our conversions, we can fill the table as follows: \( \begin{tabular}{|c|c|c|c|} \hline Radical form & Exponential form & Base & Exponent \\hline\(\sqrt[5]{25}\) & \(25^{1/5}\) & 25 & \(\frac{1}{5}\) \\((\sqrt[4]{16})^{-3}\) & \(16^{-3/4}\) & 16 & \(\frac{-3}{4}\) \& \((-27)^{2 / 3}\) & -27 & \(\frac{2}{3}\) \\(-\sqrt{\frac{9}{64}}\) & \(-\left(\frac{9}{64}\right)^{1/2}\) & \(\frac{9}{64}\) & \(\frac{1}{2}\) \\hline\end{tabular} \)

Key Concepts

Radical expressions

Radical expressions involve roots of numbers, like square roots or cube roots. The radical symbol, \( \sqrt{} \), indicates the root we are taking. In cases where there is a number in the index position, \( n \), we refer to it as the \( n \)-th root. For instance, \( \sqrt[5]{25} \) represents the fifth root of 25. This is called a radical expression.

Radical expressions come in various forms, from simple square roots such as \( \sqrt{4} \) to more complex forms like \( \sqrt[3]{27} \) or fractional expressions like \( \frac{\sqrt{7}}{2} \). They are essential in calculus and in various practical applications, especially in engineering and physics.

• Basic radical form: \( \sqrt{a} \)
• General form: \( \sqrt[n]{a} \)
• Complex form: negative and fractional radicands

Understanding radicals is crucial as they provide a bridge to exponential expressions, essential for calculations involving powers and roots.

Base and exponent identification

Identifying the base and exponent in expressions is a foundational skill in algebra. When you see something like \( a^b \), \( a \) is called the base, and \( b \) is the exponent. These components tell us how many times the base is multiplied by itself.

For example, in the expression \( 25^{1/5} \), 25 is the base and \( \frac{1}{5} \) is the exponent. This signifies the fifth root of 25.

• Base: The number being raised in a power notation (here, 25).
• Exponent: The power to which the base is raised (here, \( \frac{1}{5} \)).

Correctly identifying these parts helps in understanding and solving equations, performing calculations, and converting between radical and exponential forms. For a negative base, like in \((-27)^{2/3}\), the negative sign is part of the base making it \(-27\), whereas the exponent \(\frac{2}{3}\) indicates a compound operation of taking the cube root and then squaring the result.

Converting radical form to exponential form

Converting radicals to exponentials simplifies manipulation of expressions and is crucial for calculus and algebraic operations. The property \( \sqrt[n]{a} = a^{1/n} \) lets us express radicals using exponents.

For instance, \( \sqrt[5]{25} \) translates to the exponential form \( 25^{1/5} \). This indicates that rather than taking the fifth root, we interpret it as raising 25 to the power of \( \frac{1}{5} \). Similarly, repeating the process with nested radicals, \( (\sqrt[4]{16})^{-3} \) becomes \( 16^{1/4} \) followed by raising it to \(-3\), resulting in \( 16^{-3/4} \).

• Use \( a^{1/n} \) for \( \sqrt[n]{a} \)
• Combine with normal powers as \( (a^{m/n}) \) when necessary, meaning take the \( n \)-th root first, then raise to \( m \).

Understanding this conversion process aids students in tackling complex problems involving roots, and when dealing with equations, making transformations straightforward and more manageable.