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Chapter 6: Problem 111

Simplify each expression using the Product Property for Exponents. $$n^{y} \cdot n^{2}$$

Short Answer

Expert verified
\(n^{y + 2}\)

Step by step solution

01

Identify the Base

Both terms in the expression, \(n^{y} \text{ and } n^{2}\), share the same base, which is \(n\).
02

Apply the Product Property for Exponents

The Product Property for Exponents states that \textbf{when multiplying two expressions with the same base, add their exponents}. Therefore, we have: \(n^y \times n^2 = n^{y + 2}\)
03

Simplify the Expression

Combining the exponents gives the simplified expression: \(n^{y + 2}\)

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental concept in algebra and are used to express repeated multiplication of a number by itself. When you see an expression like \( n^y \), it means that the base \( n \) is multiplied by itself \( y \) times.
  • The number being multiplied is called the base.
  • The number of times it's multiplied by itself is called the exponent.
For example, in \( 2^3 \):
  • The base is 2.
  • The exponent is 3.
This means \( 2^3 = 2 \times 2 \times 2 = 8 \). Understanding exponents is crucial before moving forward with more complex algebraic concepts.
Algebra and the Product Property for Exponents
Algebra involves using symbols to represent numbers in equations and expressions. When working with exponents in algebra, it's essential to understand the Product Property for Exponents. The property states that when you multiply two expressions with the same base, you add their exponents.
For instance, consider the expression in the exercise: \( n^y \times n^2 \). Both terms have the same base, which is \( n \). Using the Product Property for Exponents, you add the exponents together:
\[ n^y \times n^2 = n^{y + 2} \] This property makes simplifying algebraic expressions involving exponents straightforward and manageable.
Simplifying Expressions
Simplifying expressions is a key skill in algebra that helps you reduce complex expressions into simpler forms. This process often involves using properties of exponents, such as the Product Property.
Let's revisit the exercise we solved: Simplify \( n^y \times n^2 \). We identified the base and used the Product Property to combine the exponents, resulting in the simplified expression \( n^{y+2} \).
  • First, look for like terms. In this case, both terms share the base \( n \).
  • Next, apply relevant exponent rules, such as the Product Property.
  • Finally, simplify the expression by performing any indicated operations.
With these steps, simplifying expressions becomes a clear and systematic process.

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Most popular questions from this chapter

Divide each polynomial by the monomial. $$\frac{12 q^{2}+3 q-1}{3 q}$$

Divide each polynomial by the monomial. $$\left(24 p^{2}-33 p\right) \div(-3 p)$$

Divide each polynomial by the binomial. $$\left(y^{2}+7 y+12\right) \div(y+3)$$

Divide each polynomial by the monomial. $$\frac{10 x^{2}+5 x-4}{-5 x}$$

Divide each polynomial by the monomial. $$\left(48 y^{4}-24 y^{3}\right) \div\left(-8 y^{2}\right)$$
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