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Chapter 5: Problem 87

Simplify each expression using the properties for exponents. $$ y^{a} \cdot y^{b} $$

Short Answer

Expert verified
\(y^{a+b}\)

Step by step solution

01

- Identify the base

Observe that both terms in the expression, \(y^{a} \cdot y^{b}\), have the same base, which is \(y\).
02

- Apply the product of powers property

The product of powers property states that \(a^m \cdot a^n = a^{m+n}\). In this expression, the base \(y\) is raised to two different exponents, \(a\) and \(b\).
03

- Simplify the expression

Combine the exponents by adding them together. This gives us \(y^{a+b}\).

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Key Concepts

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

simplifying expressions
Simplifying expressions is a fundamental skill in algebra that helps you reduce complex equations into simpler ones. This makes them easier to work with and understand.
When simplifying expressions involving exponents, the main goal is to combine like terms and apply the properties of exponents effectively.
If you understand these core rules, many algebra problems become straightforward.
  • Identify like terms: Look for terms with the same base. These can often be combined using exponent rules.
  • Use exponent properties: These properties help in simplifying expressions to their simplest forms.
In the original problem, the expression involves exponents with the same base, making it easier to simplify using exponent properties.
product of powers property
The product of powers property is an essential rule to know when working with exponents. This property states that when you multiply two terms with the same base, you simply add their exponents.
Mathematically, this is shown as: \(a^m \times a^n = a^{m+n}\)
In the given problem, both terms have the same base (\(y\)). Therefore, you apply the product of powers property.
  • Identify the base: In this case, the base is \(y\).
  • Combine the exponents: Add the exponents \(a\) and \(b\).
This transforms the original expression \(y^a \times y^b\) into \(y^{a+b}\). This simplified form is much easier to work with in equations and further algebraic manipulations.
algebra basics
Understanding algebra basics is crucial to mastering more advanced mathematical concepts.
Key basics include knowing the properties of exponents, how to simplify expressions, and combine like terms.
  • Variables and constants: Variables represent unknowns, and constants are known values.
  • Base and exponent: In an exponential term like \(y^a\), \(y\) is the base, and \(a\) is the exponent.
In the exercise provided, understanding the basics helps you identify that both terms in \(y^a \times y^b\) share the same base. This recognition allows you to apply the product of powers property easily and simplify the expression efficiently.

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Most popular questions from this chapter

Multiply the binomials using (a) the Distributive Property; (b) the FOIL method; (c) the Vertical Method. $$ (y+9)(y+3) $$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ (a b-4)(a b+4) $$

Find each product. $$ \left(j^{2}-8 j-27\right)-\left(j^{2}+2 j-12\right) $$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ (x y-9)(x y+9) $$

Square each binomial using the Binomial Squares Pattern. $$ \left(8 p^{3}-3\right)^{2} $$
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