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Chapter 5: Problem 21

Use the product rule to simplify each expression. $$ x^{2} \cdot x^{5} $$

Short Answer

Expert verified
The expression simplifies to \(x^7\).

Step by step solution

01

Identify the Exponents

The given problem needs the simplification of the expression \(x^2 \cdot x^5\). First, identify the exponents of the two terms. Here, \(x^2\) has an exponent of 2, and \(x^5\) has an exponent of 5.
02

Apply the Product Rule

Recall the product rule for exponents, which states that when you multiply two expressions with the same base, you add their exponents: \(x^a \cdot x^b = x^{a+b}\). Apply this rule to the given expression: \(x^2 \cdot x^5 = x^{2+5}\).
03

Simplify the Exponent

Add the exponents 2 and 5 from the previous step to simplify the expression: \(x^{2+5} = x^7\).

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Key Concepts

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

Simplification Techniques
In algebra, simplification is a crucial step in solving problems. It involves breaking down complex expressions into their simplest forms. To simplify an expression like \(x^2 \cdot x^5\), it's essential to recognize patterns and use specific rules.
  • Identify the basics: Notice that each term in the expression has the same base, \(x\) in this case.
  • Use known rules: The product rule for exponents is a powerful tool in simplifying terms with the same base.
By reducing the number of steps and rearranging terms, you not only make the problem easier to solve but also facilitate a better understanding of the mathematical principles at work. In this case, recognizing the need to apply the product rule directly simplifies the multiplication of two exponential terms into a single term with a higher exponent.
Exponents
Exponents are numerical notations used to signify repeated multiplication of a base by itself. They are essential in representing large numbers succinctly and in calculations involving powers.When you see terms like \(x^2\) or \(x^5\), the number 2 or 5 is the exponent, which tells you how many times the base \(x\) should be multiplied by itself.
  • \(x^2\): Multiply \(x\) by itself once (i.e., \(x \cdot x\)).
  • \(x^5\): Multiply \(x\) by itself four more times (i.e., \(x \cdot x \cdot x \cdot x \cdot x\)).
Applying the product rule, where \(x^a \cdot x^b = x^{a+b}\), allows you to combine exponents efficiently. In \(x^2 \cdot x^5\), it means calculating \(x^{2+5} = x^7\), simplifying the expression to a manageable form.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operators. In expressions like \(x^2 \cdot x^5\), they use exponents and multiplication to represent the operation being performed on the variable \(x\).These expressions are foundational in algebra because they can represent real-world problems or complex mathematical relationships. Understanding how to manipulate them, especially by using rules like the product rule for exponents, is crucial in algebra.To work with algebraic expressions effectively:
  • Recognize like terms: Terms with the same base variable can often be combined or simplified using exponent rules.
  • Apply appropriate rules: Use the product, quotient, or power rules to manipulate the expression.
Simplifying expressions like \(x^2 \cdot x^5\) helps in solving larger mathematical problems and aids in understanding the underlying concepts of algebra.

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Most popular questions from this chapter

Simplify each expression by performing the indicated operation. Explain how you arrived at each answer. a. \(z+3 z\) b. \(z \cdot 3 z\) c. \(-z-3 z\) d. \((-z)(-3 z)\)

Add or subtract as indicated. $$ (14 y+12)+(-3 y-5) $$

Subtract \(\left(y^{2}-9\right)\) from the sum of \(\left(3 y^{2}+y+4\right)\) and \(\left(2 y^{2}-6 y-10\right)\)

Add or subtract as indicated. $$ \left(a^{2}-a b+4 b^{2}\right)+\left(6 a^{2}+8 a b-b^{2}\right) $$

Evaluate each expression using exponential rules. Write each result in standard form. $$ \left(2.5 \times 10^{6}\right)\left(2 \times 10^{-6}\right) $$
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