Problem 8 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ 81^{1 / 4} $$

Step-by-Step Solution

Answer

The expression \(81^{1/4}\) simplifies to \(3\).

Step 1: Identify the Radical Form

The expression given is in the form of a rational exponent, specifically \(81^{1/4}\). Recall that \(a^{1/n}\) can be rewritten as the nth root of \(a\). So, we can express \(81^{1/4}\) in radical form as \(\sqrt[4]{81}\).

Step 2: Simplify the Radicand

Now, we need to simplify \(\sqrt[4]{81}\). To do this, think about what number multiplied by itself four times equals 81. Since \(3^4 = 81\), it follows that \(\sqrt[4]{81} = 3\).

Key Concepts

Rational Exponents

Rational exponents are a way to represent roots using exponential notation. Instead of using a radical sign to indicate a square root or any other root, we use a fraction as the exponent. This approach stems from the idea that exponents can be more flexible and easy to manipulate in calculations.
For example, a rational exponent \( a^{1/n} \) represents the \( n \)-th root of \( a \). If \( a \) is a real number, \( a^{1/n} = \sqrt[n]{a} \). Here, the denominator \( n \) of the fraction indicates the degree of the root we are taking.
Using rational exponents can sometimes make complex algebraic manipulation straightforward. The rules for manipulation, such as multiplying and dividing exponents, apply just as with integer exponents, which can simplify calculations significantly.

Simplification

Simplification involves reducing an expression to its simplest form. This means no further calculations or reductions can be done. Simplification is a key skill in algebra to ensure that expressions are easy to understand and work with.

• When simplifying expressions with roots and exponents, consider looking for perfect squares, cubes, or other powers within the radicand (the number inside the root).
• In the expression \( \sqrt[4]{81} \), the number 81 can be simplified because it's a perfect fourth power, i.e., \( 81 = 3^4 \). Thus, \( \sqrt[4]{81} = 3 \) because when 3 is multiplied by itself four times, the result is 81.

Simplification often involves breaking down the original expression into more basic parts and using properties of exponents and radicals to rewrite them.

Nth Root

The \( n \)-th root of a number is a fundamental concept that forms the basis of many mathematical operations. It is essentially the inverse operation of raising a number to a power.
Finding the \( n \)-th root of a number involves determining which number, when raised to the \( n \)-th power, will yield the original number.
For instance, the \( 4 \)-th root of 81 is 3 because \( 3^4 = 81 \).

• The notation \( \sqrt[n]{a} \) is used to denote the \( n \)-th root of \( a \), where \( n \) is a positive integer, and \( a \) is a real number.
• The \( n \)-th root can be understood by recognizing it as the rational exponent \( a^{1/n} \).

Working with \( n \)-th roots is essential when dealing with polynomial equations, especially when higher degree roots are involved. Understanding \( n \)-th roots can also help in solving equations that model real-world problems in science and engineering.