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Chapter 7: Problem 2

Simplify each of the following as completely as possible. $$y^{4} y^{3}$$

Short Answer

Expert verified
\( y^{7} \)

Step by step solution

01

- Identify the rule

When you multiply powers with the same base, you add their exponents. This is known as the Product of Powers Property.
02

- Apply the rule

Given the expression is \( y^{4} y^{3} \). According to the rule, add the exponents: \( 4 + 3 \).
03

- Simplify the expression

Add the exponents: \( y^{4+3} = y^{7} \).

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Key Concepts

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

Product of Powers Property
The Product of Powers Property is a fundamental rule in algebra. This property states that when you multiply two expressions with the same base, you simply add their exponents. For example, in the expression \( y^4 \: y^3 \), both terms have the base 'y'. According to this property, you add the exponents: 4 and 3.

This gives us: \[ y^{4+3} = y^7 \]. It simplifies the multiplication process significantly and is essential to understand for simplifying algebraic expressions. Remember, this rule only applies when the bases are the same.
Multiplying Exponents
Multiplying exponents is straightforward when the bases are the same. In line with the Product of Powers Property, you add the exponents.

Let's break it down: If you're given an expression like \( y^4 \: y^3 \), you identify that 'y' is the base for both terms. Then, you add the exponents: 4 and 3. Following this rule: \[ y^4 \: y^3 = y^{4+3} = y^7 \].

In more complex problems, always ensure to identify the common base and then apply the rule. It makes handling larger and more complicated expressions manageable.
Simplifying Expressions
Simplifying expressions is a crucial skill in algebra. It involves reducing an expression to its simplest form. When dealing with expressions that involve exponents, like \( y^4 \: y^3 \), using the Product of Powers Property helps achieve this.

Simplification Steps:
  • Identify the rule: Multiply powers with the same base by adding their exponents.
  • Apply the rule: For \( y^4 \: y^3 \), add the exponents: 4 + 3.
  • Simplify: The expression becomes \[ y^{4+3} = y^7 \].
By simplifying, you make the expression easier to work with and understand, ensuring it is in its most reduced form. Remember, practice helps in mastering these steps.

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Most popular questions from this chapter

perform the indicated operation and simplify. $$\text { Solve for } x: \frac{x}{3}-\frac{x}{4}=3$$

Perform the indicated operations and simplify. $$\frac{4 x^{2} y}{15 z^{3}} \cdot \frac{12 x z}{x^{2} y}$$

Perform the indicated operations and simplify. $$\frac{a}{3}-\frac{a}{2}+\frac{a}{5}$$

Convert each number into standard notation. $$0.0527 \times 10^{4}$$

What can you say about the sign of the exponent of 10 of a number written in scientific notation if the number is bigger than \(1 ?\) If the number is smaller than \(1 ?\) Why?
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