Problem 61 - Algebra and Trigonometry

Question

Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$ \sqrt[4]{(-3)^{4}} $$

Step-by-Step Solution

Answer

The value of the expression \(\sqrt[4]{(-3)^{4}}\) is \(3\).

Step 1: Evaluate the Exponent

Firstly, evaluate \((-3)^{4}\), which means \(-3\) should be multiplied by itself four times. It's important to keep in mind that when a negative number is raised to an even power, the result is a positive number.

Step 2: Take the Fourth Root

The next step is to evaluate the fourth root of the number obtained in step 1. The fourth root of a number \(n\) stands for what number, when multiplied by itself four times, gives the number \(n\). It's crucial to note that the fourth root of a positive number is always a real number.

Step 3: Solve the Problem

Now, apply these calculations step by step. Begin with evaluating \((-3)^{4}\), which equals to \(81\). Then, take the fourth root of \(81\), which results in \(3\).

Key Concepts

Exponent Rules

Understanding the rules for exponents is crucial when evaluating expressions with roots, as it allows us to simplify the numbers with which we're working. Exponents tell us how many times to use a number in a multiplication process. For example, when we see an expression like \(a^n\), it means to multiply the base \(a\) by itself \(n\) times. When the base is negative, the sign of the outcome depends on whether the exponent is even or odd.

There are some fundamental rules you should know:

• The product rule: \(a^m \times a^n = a^{m+n}\)
• The quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\), as long as \(a \eq 0\).
• The power of a power rule: \(\left(a^m\right)^n = a^{m \times n}\).
• The negative exponent rule: \(a^{-n} = \frac{1}{a^n}\) for any \(a\) not equal to zero.

These rules make it easier to simplify and evaluate expressions before taking roots, ensuring we have the simplest form of the number.

Even Powers of Negative Numbers

An important concept to remember is how even powers affect negative numbers. An even power means the exponent is a number like 2, 4, 6, and so forth, and it will result in a positive number regardless of whether the base is positive or negative. This occurs because multiplying two negative numbers together results in a positive product, and since an even number of negative factors will pair off into positive products, the overall result is positive.

For instance, with the expression \( (-3)^4 \), the base is -3 and the exponent is 4, an even number. Here's how that works out step by step:

• First pair: \( (-3) \times (-3) = 9 \)
• Second pair: \( (-3) \times (-3) = 9 \)
• Combining the pairs: \( 9 \times 9 = 81 \)

We end up with a positive 81 because four 'negatives' in total give us a positive result. Recognizing this characteristic of even powers saves us from errors and confusion when evaluating roots of these expressions.

Fourth Roots

Now, let's talk about fourth roots, which are a specific type of root that may look intimidating but follows the same logic as square roots, only they go one step further. A fourth root is indicated as \(\sqrt[4]{n}\) and asks the question: 'What number multiplied by itself four times equals \(n\)?'

The fourth root of a number can be positive, negative, or even zero. If the number \(n\) is positive, the fourth root will have two real values, one positive and one negative, because both \(a \times a \times a \times a\) and \( (-a) \times (-a) \times (-a) \times (-a)\) will result in a positive \(n\). However, it is standard to consider the principal (positive) root when looking for the fourth root. Here's an example:

• To find \(\sqrt[4]{81}\), we determine the number that multiplied by itself four times equals 81. In this case, it's 3, because \(3^4 = 81\).

When working with fourth roots, the concept of even powers of negative numbers is essential, as it guarantees that if the underneath expression is an even power of a negative number, the result will be a real number, and thus the fourth root will also be real.