Problem 74 - Algebra 2

Question

Simplify each radical expression. Use absolute value bars where they are needed. $$ \sqrt[4]{x^{4}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[4]{x^{4}} \) is \( |x| \)

Step 1: Identify the expression inside the radical

The given expression is \( \sqrt[4]{x^{4}} \). The expression inside the 4th root is \( x^4 \)

Step 2: Apply the rule of exponents

According to the rules of exponents, when a number or variable is raised to a power and that result is then under a root of the same degree, the result is the absolute value of the original number or variable. So, the result should be \( |x| \)

Step 3: Simplify the expression

Substitute \( x^4 \) with \( |x| \) in the expression. The simplified expression is \( |x| \)

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of the same number or variable. For example, if we say \( x^4 \), it means \( x \times x \times x \times x \). Here, \( 4 \) is the exponent, telling us how many times to multiply \( x \) by itself. Exponents are a fundamental concept in algebra, appearing often in equations and expressions.
There are some key rules of exponents that simplify calculations:

• Product Rule: \( a^m \times a^n = a^{m+n} \)
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power Rule: \( (a^m)^n = a^{m \times n} \)

These rules allow us to manipulate expressions with exponents effectively.
When using radicals, understanding how they relate to exponents is crucial. An \( n \)-th root of a number is the same as raising that number to a fraction, like \( a^{1/n} \). Therefore, \( \sqrt[4]{x^4} \) can also be written as \( (x^4)^{1/4} \), simplifying to \( x^{4/4} = x^1 = x \), but with absolute value consideration.

Absolute Value

Absolute value is a concept that measures the distance of a number from zero on a number line, regardless of direction. It is always non-negative, which is why it appears as \( |x| \) when simplifying certain expressions. This is particularly important when dealing with even roots because negative numbers cannot have even roots in the real number system.
For any real number \( x \), the absolute value is defined as:

• \( |x| = x \), if \( x \geq 0 \)
• \( |x| = -x \), if \( x < 0 \)

Thus, \( |x| \) ensures the result is always positive or zero. When simplifying expressions like \( \sqrt[4]{x^4} \), we use absolute value because \( x \) could be either positive or negative. If \( x \) were negative, the expression would still evaluate to a positive result \( |x| \), maintaining integrity in the solution.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, or any \( n \)-th root. They are used to indicate the inverse operation of raising a number to a power. For example, \( \sqrt{x} \) represents the square root of \( x \), while \( \sqrt[4]{x^4} \) stands for the fourth root of \( x^4 \).
Simplifying radical expressions often involves rewriting them using fractional exponents. This can make calculations easier and expressions simpler to handle:

• The expression \( \sqrt[n]{x^m} \) can be written as \( x^{m/n} \).
• If \( m \) is equal to \( n \), like in \( x^{4/4} \), it simplifies to \( x^1 = x \).

Utilizing the absolute value in such cases is critical because it guarantees that the result of an even root stays non-negative. Without using absolute values, the result could wrongly reflect a negative quantity when no real negative counterparts exist, especially for even radicals.