Problem 5 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression using radicals. Radical expression \(\quad\) Exponential expression \(\sqrt[5]{5^{3}}\) \(\quad\) ______

Step-by-Step Solution

Answer

\(5^{\frac{3}{5}}\)

Step 1: Identify the Base Expression

The given radical expression is \(\sqrt[5]{5^3}\). Here, the base expression is \(5^3\), which is under the fifth root.

Step 2: Understand the Radical to Exponential Conversion

In mathematical terms, \(\sqrt[n]{a^m}\) can be expressed as \(a^{\frac{m}{n}}\). This means that the radical notation can be converted to an exponent by changing the expression to a fractional exponent.

Step 3: Substitute Values in the Formula

Apply the conversion rule to \(\sqrt[5]{5^3}\). This can be written as \(5^{\frac{3}{5}}\) since the index of the root \(n = 5\) and the exponent under the root \(m = 3\).

Key Concepts

Understanding Radical Expressions

Radical expressions involve roots of numbers or variables. When you see a square root or higher degree root, it's expressed with a radical symbol \(\sqrt{}\). For example, \(\sqrt{9}\) is the square root of 9. The number under the radical symbol is called the radicand. When the radical is more than a square root, it will have a small number outside the radical symbol, indicating the degree of the root, like \(\sqrt[3]{8}\), which is the cube root of 8.
Radical expressions are useful in simplifying equations, especially when dealing with areas, volumes, and growth calculations. Higher degree roots, such as cube roots, quartic roots, etc., are often used to express complexities in various algebraic contexts.

• The symbol \(\sqrt{}\) represents the radical.
• The radicand is the number inside the radical.
• The index is the small number that indicates the degree of the root.

Exploring Exponential Expressions

Exponential expressions are mathematical expressions wherein a number or variable, called the base, is raised to a power or an exponent. The exponent tells us how many times the base is multiplied by itself. A simple exponential expression is \(5^3\), which equals 5 multiplied by itself twice more (\(5 \times 5 \times 5=125\)).
These expressions are crucial in fields such as population growth, finance, and computing, where repeated multiplication is commonplace. Exponential expressions help us simplify and manage calculations involving large numbers effectively.

• The base is the number that is multiplied.
• The exponent represents how many times the base is used as a factor.
• Exponential expressions can also be negative or fractional, e.g., \(5^{-2}\) or \(5^{\frac{1}{3}}\).

Conversion Between Radicals and Exponents

Converting between radical and exponential forms is a key algebraic technique. It simplifies working with expressions that involve roots. Radicals can be written as exponents with fractional powers, where the numerator represents the power of the radicand and the denominator is the index of the root.
For example, \(\sqrt[5]{5^3}\) can be expressed as \(5^{\frac{3}{5}}\). This conversion uses the formula: \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\). Applying this to complex problems makes them easier to solve, as exponent rules are often simpler to work with than radicals.

• Identify the base and the index of the root in the radical.
• Apply the formula \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\).
• Use this conversion to handle equations involving both roots and powers effectively.