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Chapter 4: Problem 65

Simplify. $$x^{5} x^{5}$$

Short Answer

Expert verified
The simplified form of \(x^{5} x^{5}\) is \(x^{10}\).

Step by step solution

01

Identify the bases and the exponents

The base in this case is \(x\) and each base has the exponent 5.
02

Apply the rule of multiplication for exponents

According to the rule of multiplication for exponents, you add the exponents when the bases are the same. Here, you add 5 and 5 to get 10.
03

Write the final answer

The simplified form of \(x^{5} x^{5}\) is \(x^{10}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. Variables represent unknown values and can stand for numbers that can vary. For instance, in the expression (x^5 x^5can be considered an algebraic expression, where x is the variable and the number 5 is an exponent indicating the power to which the variable is raised.
Simplifying Exponents
Simplifying exponents, also known as rationalizing powers, is the process of reducing expressions with exponents to their simplest form. This is a fundamental concept in algebra that helps make complex problems more manageable. To simplify exponents, we can use exponent rules like the product rule or the power rule. In the given exercise, the expression (x^5 x^5) simplifies to (x^{10}) by adding the exponents since the bases are the same.
Exponent Rules
Exponent rules are guidelines that describe how to handle exponents in mathematical expressions. One such rule is the product rule for exponents, which states that when multiplying two expressions with the same base, you add their exponents. In the exercise (x^5 x^5), we apply the product rule to get a single exponent by adding them, resulting in the simplified expression (x^{10}). Understanding each of these exponent rules is crucial for simplifying algebraic expressions efficiently.

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