Problem 49 - College Algebra

Question

Simplify each exponential expression $$ \frac{8 x^{20}}{2 x^{4}} $$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \(4x^{16}\).

Step 1: Identify the Base

In this exercise, the base is \(x\). It occurs in both the numerator and the denominator.

Step 2: Apply the Exponent Rules

The rule to apply here is \(x^a / x^b = x^{a-b}\). This rule states that to divide two exponents with the same base, subtract the exponent of the base in the denominator from the exponent of the base in the numerator. In this case, the exponents are 20 and 4.

Step 3: Simplify the Numerical Coefficient

The numerical coefficient is 8/2. Simplify this fraction to get the numerical part of the answer.

Step 4: Combine the Results

Combine the results from step 2 and step 3 to get the final simplified form of the exponential expression.

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when working with algebraic expressions involving powers. These rules, also known as the laws of exponents, allow us to manipulate expressions and perform calculations more efficiently. For example, when you have the same base in both the numerator and the denominator, as seen in the exercise \(\frac{8 x^{20}}{2 x^{4}}\), the rule to apply is \(x^a / x^b = x^{a-b}\). This particular rule tells us that to divide exponents with the same base, we simply subtract the exponent in the denominator from the exponent in the numerator.
Other vital exponent rules include:

• \(x^a \cdot x^b = x^{a+b}\), which is used for multiplying exponents with the same base.
• \((x^a)^b = x^{a\times b}\), for an exponent raised to another exponent.
• \(x^0 = 1\), which states that any non-zero base raised to the power of zero equals one.
• \(x^{-a} = \frac{1}{x^a}\), which is how you deal with negative exponents.

By understanding and applying these rules, you can simplify complex expressions and solve algebraic problems more effectively.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In the context of the given exercise, \(\frac{8 x^{20}}{2 x^{4}}\), the expression includes a numerical coefficient (8 and 2), a variable (\