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Chapter 6: Problem 112

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

Short Answer

Expert verified
y^{a+b}

Step by step solution

01

- Identify the bases and exponents

Determine the bases and the exponents in the given expression. In this case, both bases are the same (y), with exponents a and b.
02

- Apply the Product Property of Exponents

The Product Property of Exponents states that when you multiply two expressions with the same base, you add the exponents. That is, for any real numbers a and b and any non-zero base y, \[ y^{a} \times y^{b} = y^{a+b} \]
03

- Simplify the expression

Using the property identified in Step 2, add the exponents a and b: \[ y^{a} \times y^{b} = y^{a+b} \]

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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 task in algebra. Simplification helps in reducing complex expressions into a more manageable or interpretable form.

In the given exercise, you are asked to simplify the expression using the Product Property for Exponents. This involves recognizing and applying algebraic rules to combine or reduce terms.

The Product Property of Exponents is a handy tool in simplifying expressions involving powers. Recognizing patterns and properties, like identifying the same base, plays a crucial role.
exponents
Exponents are a way to represent repeated multiplication of a number. For instance, \(y^a\) means y is multiplied by itself a times. Exponents are essential in various areas of math, including algebra, calculus, and more.

In our example, we have two terms with the same base, y, raised to different exponents, a and b. The challenge is to simplify this product of exponents into a single expression.

Applying the Product Property of Exponents simplifies this as \(y^a \times y^b = y^{a+b}\). By adding the exponents, the expression becomes easier to work with.
algebraic properties
Algebraic properties help us understand how numbers and variables interact with one another. One key algebraic property is the Product Property of Exponents, which we used in the exercise.

This property tells us that when multiplying like bases, you can add the exponents: \[ y^a \times y^b = y^{a+b} \]. Recognizing and applying these properties simplifies the expressions and solves problems efficiently.

Being familiar with such properties not only makes algebraic operations easier but also lays the foundation for more advanced mathematical concepts.

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Most popular questions from this chapter

Divide each polynomial by the binomial. $$\left(x^{2}+2 x-30\right) \div(x-5)$$

Divide each polynomial by the binomial. $$\left(x^{2}-3 x-10\right) \div(x+2)$$

Divide each polynomial by the monomial. $$\frac{35 a^{4}+65 a^{2}}{-5}$$

Divide each polynomial by the monomial. $$\frac{52 p^{5} q^{4}+36 p^{4} q^{3}-64 p^{3} q^{2}}{4 p^{2} q}$$

Divide each polynomial by the monomial. $$\frac{72 r^{5} s^{2}+132 r^{4} s^{3}-96 r^{3} s^{5}}{12 r^{2} s^{2}}$$
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