Problem 11 - Algebra 2

Question

Find all the real fourth roots of each number. $$ 0.0081 $$

Step-by-Step Solution

Answer

\( \pm\frac{3}{10} \)

Step 1: Identify the Number

Recognize that the number given is 0.0081, and our task is to find all real fourth roots of this number.

Step 2: Convert to Radical Form

To find the fourth root of a number, we represent the number in radical form. The fourth root of 0.0081 is written as \( \sqrt[4]{0.0081} \).

Step 3: Convert to Fractional Exponent

Converting the radical to a fractional exponent, the fourth root of 0.0081 is represented as \( 0.0081^{1/4} \).

Step 4: Simplify the Decimal

Express 0.0081 as a fraction: \( 0.0081 = \frac{81}{10000} \). The fractional exponent becomes \( (\frac{81}{10000})^{1/4} \).

Step 5: Apply the Exponent to Numerator and Denominator

Apply the fourth root to both the numerator and the denominator separately: \( \sqrt[4]{81} = 3 \) and \( \sqrt[4]{10000} = 10 \).

Step 6: Write the Fourth Root

Combine the results to express the fourth root of the original number as \( \frac{3}{10} \).

Step 7: Consider All Real Fourth Roots

Since all even roots have positive and negative solutions, there are two real fourth roots of 0.0081, which are \( \frac{3}{10} \) and \( -\frac{3}{10} \).

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions containing a radical symbol (√), which indicates the root of a number. The most common radical expression is the square root, but we can extend it to cube roots, fourth roots, and beyond. For example, the fourth root of 16 would be written as \( \sqrt[4]{16} \) and equals 2, because 2 raised to the 4th power, or \( 2^4 \), is 16.

Fractional Exponents

Fractional exponents are an alternative way to express radicals. The expression \( \sqrt[4]{0.0081} \) is equivalent to \( 0.0081^{1/4} \). The denominator of the fraction indicates the root: in this case, the 4th root. The numerator represents the power to which the number inside the radical is raised. Fractional exponents follow the same arithmetic rules as other exponents and are often easier to work with, especially when using a calculator or when further algebraic manipulation is necessary.

Simplifying Decimals

When dealing with decimals in radical expressions, it's often helpful to convert them into fractions to simplify. The decimal 0.0081 can be written as the fraction \( \frac{81}{10000} \). This can make it easier to see patterns and work with the numbers. When expressed as a fraction, the approach to find the fourth root of each part (numerator and denominator) separately becomes more straightforward and can be handled just like integer numbers.

Solving Radical Equations

Solving radical equations involves isolating the radical on one side of the equation and then eliminating it. For example, if we want to find the fourth root of a number, we start with an equation like \( x^4 = 0.0081 \) and solve for \( x \). To eliminate the fourth root, we raise both sides of the equation to the fourth power. However, we must always remember that raising both sides of an equation to an even power can introduce an extraneous solution, so we must check our solutions in the original equation.

In the case of the number 0.0081, we found two real fourth roots: \( \frac{3}{10} \) and \( -\frac{3}{10} \). This is because any positive or negative number raised to the fourth power will give a positive result, hence why even roots can have both positive and negative solutions.