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Chapter 7: Problem 8

Simplify the following exponential expressions. $$ -7 x^{2}\left(5 x^{3}\right) $$

Short Answer

Expert verified
-35\(x^5\)

Step by step solution

01

Distribute the Constants

Distribute the constants outside the exponents. In this case, multiply -7 by 5 to get -35.
02

Apply the Product Rule for Exponents

Use the product rule for exponents, which states that when multiplying like bases, add the exponents (i.e., \(a^m * a^n = a^{(m+n)}\) ). Here, add the exponents of \(x\): \(x^2 * x^3 = x^{(2+3)} = x^5\).
03

Combine the Results

Combine the results from steps 1 and 2. The simplified expression is -35 multiplied by \(x^5\), resulting in -35\(x^5\).

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Key Concepts

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

product rule for exponents
The product rule for exponents is a fundamental concept in algebra. It states that when you multiply two exponential expressions with the same base, you simply add the exponents. For example, with the base 'a', if we have two expressions:
  • \(a^m\)
  • \(a^n\)
Multiplying them together uses the rule: a^m * a^n = a^{(m+n)}. This rule simplifies the process of dealing with exponential expressions and helps streamline calculations. In our exercise, multiplying \(x^2\) and \(x^3\) together results in \(x^{(2+3)} = x^5\).
simplifying expressions
Simplifying expressions is the process of reducing an algebraic expression to its most basic form. This often involves combining like terms and using rules such as the product rule for exponents. For example, in the exercise given:
  • Distribute constants: -7 multiplied by 5 gives -35.
  • Apply product rule: \(x^2 * x^3\) simplifies to \(x^5\) by adding the exponents.
Combining these steps, the simplified expression is -35\(x^5\). Simplification makes complex equations easier to work with and is essential for solving algebraic problems efficiently.
basic algebra
Basic algebra lays the foundation for more advanced mathematical concepts. It includes understanding operations with variables, constants, and the basic rules that govern their interactions. Key components include:
  • Variables and Constants: Variables like 'x' and constants like -7 and 5 in our exercise.
  • Arithmetic Operations: Addition, subtraction, multiplication, and division.
  • Algebraic Rules: Rules like the product rule for exponents help us manipulate expressions effectively.
In our example, combining these basic principles allowed us to simplify the expression -7\(x^2\) (5\(x^3\)) to -35\(x^5\). Understanding these fundamental concepts is essential for progressing in mathematics.

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Most popular questions from this chapter

Simplify the following exponential expressions. $$ 2^{3} 2^{3} $$

Compute the value of each of the following exponential expressions. $$ 5(-2)^{2} $$

Find the reciprocal of each of the following numbers. $$ \frac{4}{3} $$

Simplify each of the following exponential expressions. $$ \left(5^{2}\right)^{3} $$

Find the reciprocal of each of the following numbers. $$ -\frac{2}{3} $$
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