Problem 36 - Intermediate Algebra

Question

Find each root. Assume that all variables represent nonnegative real numbers. $$ \sqrt[4]{x^{20}} $$

Step-by-Step Solution

Answer

The root is \( x^5 \).

Step 1: Recognize the Root Formula

The expression is \( \sqrt[4]{x^{20}} \), which denotes the fourth root of \( x^{20} \). A general way to simplify \( \sqrt[n]{a^m} \) is to express it as \( a^{m/n} \).

Step 2: Apply the Power of a Power Rule

Using the formula \( \sqrt[n]{a^m} = a^{m/n} \), we have: \( \sqrt[4]{x^{20}} = x^{20/4} \).

Step 3: Simplify the Exponent

Calculate \( 20/4 \) to simplify the expression. \( 20/4 = 5 \). Therefore, \( x^{20/4} = x^5 \).

Step 4: Interpret the Result

The expression \( \sqrt[4]{x^{20}} \) simplifies to \( x^5 \) under the assumption that \( x \) is a nonnegative real number.

Key Concepts

Root Formula

The root formula is a powerful concept in mathematics that helps us simplify expressions involving roots. When we see an expression like \( \sqrt[n]{a^m} \), it means finding the \( n \)-th root of \( a^m \). To simplify this, we use the root formula, which allows us to rewrite it in terms of exponents as \( a^{m/n} \). Understanding this transformation is essential:

• \( \sqrt{a} \) can be expressed as \( a^{1/2} \).
• \( \sqrt[n]{a} \) is written as \( a^{1/n} \).
• In the case of multiple exponents, \( \sqrt[n]{a^m} = a^{m/n} \).

These transformations help to solve complex problems by making the manipulations of roots more straightforward and systematic. This makes it easier to work on expressions involving not just numbers but variables too.

Power of a Power Rule

The power of a power rule is used when you have an exponent raised to another exponent. This rule can be expressed as \((a^m)^n = a^{m \cdot n}\). This concept is especially useful when dealing with radical expressions where exponents are involved. Here's how it applies:

• In the example \( \sqrt[4]{x^{20}} \), replacing the radical with exponential notation gives us \( x^{20/4} \).
• This simplification uses the property that raising a power to another power multiplies the exponents, hence \( x^5 = x^{20/4} \).

Understanding this rule simplifies expressions and allows us to handle more complex algebraic manipulations easily. Keep in mind, the power of a power rule is key to mastering problems involving nested exponents.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. The expression \( a^m \) is read as "\( a \) raised to the power of \( m \)", meaning \( a \) is multiplied by itself \( m \) times. Here's what you need to know:

• \( a^1 = a \), which indicates that any number to the power of one is itself.
• \( a^0 = 1 \) for any nonzero \( a \); this is because any nonzero number raised to zero is one.
• Negative exponents represent the reciprocal: \( a^{-m} = \frac{1}{a^m} \).

In the problem \( \sqrt[4]{x^{20}} \), simplification involves understanding that \( x^{20/4} = x^5 \) is a direct application of exponent rules. This means that instead of dealing with the root directly, we converted it using the exponents, which streamline calculations and algebraic operations.