Problem 7 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression using radicals. $$ \begin{array}{l}{\text { Radical expression }} \\ {\qquad \frac{1}{\sqrt{5}}}\end{array} $$

Step-by-Step Solution

Answer

The expression \( \frac{1}{\sqrt{5}} \) is written as \( 5^{-1/2} \) using exponents.

Step 1: Understanding the Problem

The task is to convert the given radical expression \( \frac{1}{\sqrt{5}} \) into an exponential expression. Radicals and exponents are closely related; a square root can also be expressed as a power.

Step 2: Express the Radical as an Exponent

Recall that the square root of a number \( a \) can be written as \( a^{1/2} \). Therefore, \( \sqrt{5} \) can be rewritten as \( 5^{1/2} \).

Step 3: Apply the Exponent Rule

Since the expression is \( \frac{1}{\sqrt{5}} \), we rewrite it with exponents as \( \frac{1}{5^{1/2}} \). Using the exponent rule \( a^{-b} = \frac{1}{a^b} \), this becomes \( 5^{-1/2} \).{

Step 4: Rewrite the Exponential Expression

The final expression, \( 5^{-1/2} \), is the exponential form of the given radical expression \( \frac{1}{\sqrt{5}} \).

Key Concepts

Radical Expressions

Radical expressions involve roots of numbers, and they are often represented using the radical symbol \( \sqrt{} \). A common radical is the square root, denoting the value that must be multiplied by itself to obtain the original number. For example, in the expression \( \sqrt{9} \), the square root is 3, because \( 3 \times 3 = 9 \). In our exercise, the radical expression is \( \frac{1}{\sqrt{5}} \).
Radical expressions can be simplified or converted into exponential form to ease calculations or further manipulation. Understanding this relationship between radicals and exponents is key in algebra and higher-level math.

• **Understanding Radicals**: Recognize \( \sqrt{a} \) as a radical expression representing the square root of \( a \).
• **Application**: Use radical forms to solve roots or simplify expressions in calculations where roots are involved.

Exponential Expressions

Exponential expressions revolve around the concept of numbers raised to a power, which are written in the form \( a^b \) where \( a \) is the base and \( b \) is the exponent. This format allows us to work quickly with large numbers or fractions, as they provide a concise representation of repeated multiplication or division.
In the context of our exercise, converting our given radical expression to an exponential form involves rewriting \( \frac{1}{\sqrt{5}} \) using exponents, which results in \( 5^{-1/2} \).

• **Representation**: Express roots as fractional exponents in exponential expressions.
• **Exponent Rules**: Use rules like \( a^{-b} = \frac{1}{a^b} \) to convert expressions efficiently.

Conversion of Radicals

The conversion process between radicals and exponential expressions involves understanding that roots can be expressed with fractional exponents. This transformation depends on the root type involved. For instance, square roots are expressed as \( a^{1/2} \), cube roots as \( a^{1/3} \), etc.
In our exercise, the square root \( \sqrt{5} \) was converted to \( 5^{1/2} \), which allowed the expression \( \frac{1}{\sqrt{5}} \) to be reformulated as \( 5^{-1/2} \) using exponent rules. This conversion makes complex expressions easier to handle, especially when solving equations.

• **Fractional Exponents**: Know that \( \sqrt{a} = a^{1/2} \).
• **Simplifying**: Turn roots into exponents to simplify or solve algebraic expressions.
  

When converting, be attentive to maintaining the integrity of the expression during transformation.

Square Roots as Exponents

Square roots, a specific type of root, are especially common in algebra. They indicate what value, when multiplied by itself, will produce the original number. When written as an exponent, the square root \( \sqrt{a} \) becomes \( a^{1/2} \). This equivalence is primarily what aids in converting radical expressions to exponential ones.
In practice, switching to the exponent form \( a^{1/2} \) benefits algebraic operations that require manipulation of powers, integer operations, and simplification of fractional or irrational numbers.

• **Exponential Form**: Replace \( \sqrt{a} \) with \( a^{1/2} \) for clarity and simplicity in computations.
• **Manipulating Powers**: Adapt your expression mechanism to tackle complicated equations or calculus problems more smoothly.

Understanding this conversion is crucial not only in solving problems like our exercise but also in broader applications throughout mathematics.