Problem 61 - College Algebra

Question

Evaluate each expression or indicate that the root is not a real number. $$\sqrt[4]{(-3)^{4}}$$

Step-by-Step Solution

Answer

The solution to \( \sqrt[4]{(-3)^{4}} \) is 3.

Step 1: Evaluate the Exponent

First, evaluate the exponent \((-3)^{4}\) which is equal to \(81\). This is done by multiplying -3 by itself 4 times.

Step 2: Calculate the Fourth Root

Next, calculate the fourth root of 81, which is \(3\). This is because \(3^{4} = 81\).

Step 3: Final Result

Finally, combining all the steps we find that the value of \( \sqrt[4]{(-3)^{4}} \) is 3.

Key Concepts

Understanding Exponents

An exponent refers to the number of times a number, known as the base, is multiplied by itself. When you see an expression like \((-3)^{4}\), it indicates that -3 is being multiplied by itself four times:

• The base here is -3.
• The exponent is 4.

This leads to the calculation \((-3) \times (-3) \times (-3) \times (-3)\). Each pair of -3s multiplied results in 9 (since \((-3) \times (-3) = 9\)), and doing this once more results in 81. Therefore, \((-3)^4 = 81\).
Exponents make it easy to express repeated multiplication succinctly. They are key in summarizing expansive calculations neatly.

Exploring Fourth Roots

A fourth root is a number that, when multiplied by itself four times, equals the original number. If you start with 81 and want to find its fourth root, you are trying to determine what number makes \(x^4 = 81\) true.

• The expression is \(\sqrt[4]{81}\).
• The answer is 3, because when 3 is multiplied by itself four times \((3 \times 3 \times 3 \times 3)\), you get 81.

Calculating fourth roots involves working backwards from exponents. Sometimes, roots can be difficult to determine if the number is not a perfect power of four, requiring estimation or special techniques.
Fourth roots are a type of radical, a concept involving the inverse operation of exponentiation, which is crucial in many branches of mathematics.

Real Numbers and Their Roots

Real numbers include all the numbers that can be found on the number line. This encompasses a wide range of numbers, including whole numbers, integers, fractions, and decimals. Importantly, real numbers are significant in evaluating expressions involving roots and exponents.

• A real number can be positive, negative, or zero.
• Roots of real numbers might not always be real; for instance, even roots of negative numbers are not real in the set of real numbers.
• However, when taking even roots of positive numbers—as with finding the fourth root of 81—the result will be a real number, in this case, 3.

Understanding real numbers and how they interact with radical expressions helps in determining whether expressions yield real results or not, providing clarity when solving mathematical problems.