Problem 42 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt[4]{3} $$

Step-by-Step Solution

Answer

The expression \( \sqrt[4]{3} \) is written as \( 3^{1/4} \) in simplest form.

Step 1: Recognize the Radical

The expression given is the fourth root of 3, which is written as \( \sqrt[4]{3} \). Here, the number 4 indicates the degree of the root.

Step 2: Rewrite the Radical as an Exponent

To express the radical using exponents, we use the property \( \sqrt[n]{a} = a^{1/n} \). For this problem, \( n = 4 \) and \( a = 3 \), so we rewrite the expression as \( 3^{1/4} \).

Step 3: Simplify the Exponent

The given expression, \( 3^{1/4} \), is already in its simplest form because there are no factors to simplify further. The variable is raised to a positive fractional exponent.

Key Concepts

Exponents

Exponents are a way to express how many times a number, known as the base, is multiplied by itself. The expression can be represented as \( a^n \), where \( a \) is the base and \( n \) is the exponent or power. Here's what happens:

• If the exponent is positive, \( a^n \) means multiply \( a \) by itself \( n \) times.
• If the exponent is zero, \( a^0 \) equals 1, based on the rule of exponents.
• If the exponent is negative, \( a^{-n} \) entails the reciprocal otherwise given by \( \frac{1}{a^n} \).

Exponents are particularly useful for simplifying large numbers, maintaining scalability, and handling complex equations. Exponents can be seen everywhere, from scientific notation to expressing square roots using fractional exponents.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest form. A radical involves roots such as square roots, cube roots, and so on. To simplify a radical:

• Identify any perfect squares (or cubes, etc.) that divide evenly into the number under the radical.
• Rewrite the number under the radical as the product of the perfect square and another number.
• Take the square root (or other root) of the perfect square out of the radical.

For example, what if you have \( \sqrt{18} \)? Since \( 18 \) can be rewritten as \( 9 \times 2 \), and we know \( \sqrt{9} = 3 \), \( \sqrt{18} \) simplifies further to \( 3\sqrt{2} \). This process helps in reducing complex numbers, making them easier to work with in equations.

Fractional Exponents

Fractional exponents provide a link between radicals and exponents. When you see an expression like \( a^{m/n} \), it corresponds to the \( n \)-th root of \( a \) raised to the \( m \)-th power, or \( (\sqrt[n]{a})^m \).

• The numerator, \( m \), denotes a power to which the base is raised.
• The denominator, \( n \), signifies the index of the root.

To convert a radical expression to a fractional exponent, consider the depth of the radical. For instance, \( \sqrt[4]{3} \) becomes \( 3^{1/4} \) using the rule \( \sqrt[n]{a} = a^{1/n} \). Implementing fractional exponents provides more efficient computation and easier multiplication or division in equations involving roots or powers.