Problem 61 - College Algebra Essentials

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 final result of the expression \(\sqrt[4]{(-3)^{4}}\) is 3

Step 1: Raising -3 to the power of 4

First, we raise -3 to the power of 4, which is \((-3)^{4}=81.\) This happens because when we raise a negative number to an even power, the result is always a positive number.

Step 2: Find the fourth root

Then calculate the fourth root of 81, which is \(\sqrt[4]{81}= 3\). This is because 3 raised to the power of 4 gives 81, so 3 is the fourth root of 81.

Key Concepts

Rational Exponents

Rational exponents are a way to express roots as exponents. They follow the general rule that

• for a number or expression x, raising it to a rational exponent of the form \( \frac{m}{n} \) signifies taking the nth root of x and then raising the result to the mth power.
• This can be written as \( x^{\frac{m}{n}} = (\sqrt[n]{x})^m \).

Using rational exponents provides a unified way to deal with both roots and powers, making calculations simpler.
By converting roots to exponents and vice versa, we can handle expressions more flexibly and often faster, especially in algebra and calculus.
The letters and numbers in rational exponents convey crucial information:

• The denominator indicates the root's degree
• The numerator indicates the power

Even Powers

Even powers refer to any expression in which an exponent is an even number, such as 2, 4, 6, etc.
Raising any real number to an even power will result in a positive value.

• For example, \( (-3)^4 = (-3) imes (-3) imes (-3) imes (-3) = 81 \).
• When a negative number is used as the base, the repeated multiplication will cancel out the negative signs.
• This underscores a fundamental property: Even powers always produce non-negative results for real numbers.

Working with even powers can simplify our tasks in identifying the nature (positive/negative) of expressions, which is particularly useful when evaluating radical expressions.

Real Numbers

Real numbers encompass all numbers you can plot along an infinite line without gaps.
This line stretches from negative infinity through zero to positive infinity.
Real numbers include:

• Every integer (e.g., -3, 0, 4)
• Fractions and decimals (e.g., \( \frac{1}{2} \, \text{and} \, 3.75 \) )

Real numbers are crucial in evaluating whether certain roots are valid.
When working with radical expressions and roots, knowing the nature of real numbers helps in determining solution possibilities.
If an expression's solution falls outside these boundaries, it might not represent a real number, prompting caution and re-evaluation in mathematical exercises.

Fourth Root

The fourth root of a number seeks a value which, when multiplied by itself four consecutive times, returns the original number.
For example,

• We know that 3 raised to the power of 4 results in 81.
• This relationship tells us that 3 is the fourth root of 81.

The fourth root is expressed using an index of 4 in the radical symbol, as in \(\sqrt[4]{81}\).
This specific root type connects closely to rational exponents; it can be expressed as \( 81^{\frac{1}{4}} \), making it versatile and applicable in various mathematical approaches.
Understanding the fourth root helps in simplifying complex radical expressions and enables us to solve equations involving higher-degree equations effectively.