Problem 1 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression using radicals. Radical expression \(\quad\) Exponential expression \(\frac{1}{\sqrt{5}}\) \(\quad\) ______

Step-by-Step Solution

Answer

The expression is \( 5^{-1/2} \).

Step 1: Understanding Radicals and Exponents

To convert radicals to exponents and vice versa, we use the relationship between them. The square root of a number can be written as the number raised to the power of one-half. For example, \( \sqrt{x} = x^{1/2} \). This is a fundamental principle to keep in mind when switching between the two forms.

Step 2: Express the Given Radical as an Exponent

The given radical expression is \( \frac{1}{\sqrt{5}} \). To express this using exponents, rewrite the square root as an exponent: \( \sqrt{5} = 5^{1/2} \). Therefore, \( \frac{1}{\sqrt{5}} = 5^{-1/2} \). This conversion uses the property that \( 1/a^b = a^{-b} \).

Step 3: Ensure Correct Exponential Notation

Ensure the expression \( 5^{-1/2} \) correctly represents \( \frac{1}{\sqrt{5}} \). The exponent \( -1/2 \) indicates both the reciprocal and the square root, matching the original expression.

Key Concepts

Exponents

Understanding exponents is key to mastering all things related to powers in mathematics. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \(5^3\), the base is 5, and the exponent is 3, meaning \(5 \times 5 \times 5\).
Exponents can be positive, negative, or fractional:

• Positive exponents show simple repeated multiplication.
• Negative exponents signify the reciprocal of the base raised to the corresponding positive power. For instance, \(5^{-1} = \frac{1}{5}\).
• Fractional exponents denote roots, such as \(5^{1/2} = \sqrt{5}\).

By mastering these concepts, you can easily switch between radical and exponential expressions.

Square Roots

Square roots are a fundamental concept in algebra with distinct characteristics and applications. They are the opposite of squaring a number. For instance, since \(3^2 = 9\), it follows that \(\sqrt{9} = 3\).
This fundamental relationship is essential when working with expressions involving radicals. Certain properties associated with square roots include:

• A square root of a number \(x\) is represented as \(\sqrt{x}\). It's the number that, when multiplied by itself, gives \(x\).
• The square root of a number \(x\) can also be written in exponential form as \(x^{1/2}\).
• Simplifying expressions with square roots enables us to distribute, combine, or rearrange factors within equations or inequalities.

A solid understanding of square roots will significantly aid in comprehending more complex algebraic and geometric situations.

Exponential Expressions

Exponential expressions involve raising numbers or variables to a power, making them a compact way to express repeated multiplication. They form the basis of more complex mathematical phenomena, like exponential growth or decay.
Some key aspects of exponential expressions include:

• The base which is the number being multiplied.
• The exponent that indicates the number or variable's multiplying times or how many times it's used as a factor.
• The rules for simplifying exponential expressions, such as \(a^m \cdot a^n = a^{m+n}\) or \(\left(a^m\right)^n = a^{m\cdot n}\).
• Fractional exponents are often converted to radical expressions for simplification; for example, an expression like \(x^{3/2}\) can be written as \(x^{3/2} = \sqrt{x^3}\).

Understanding how to manipulate exponential expressions is fundamental in solving equations involving exponential growth, logarithms, and even calculus.