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Chapter 9: Problem 15

For Exercises,$ simplify. $$\frac{y^{7}}{y^{3}}$$

Short Answer

Expert verified
The simplified form of \( \frac{y^{7}}{y^{3}} \) is \(y^{4}\).

Step by step solution

01

Identify the Problem

Here, we have a division operation where the base for both numerator and denominator are the same (which is 'y'), and we need to simplify the expression as much as possible.
02

Apply the Rule of Exponents

According to the rule of exponents, when we divide expressions with the same base, the exponent of the result is the difference of the exponents in the initial terms. Therefore, we subtract the exponent of the denominator from the exponent of the numerator, which gives us \(y^{7-3}\).
03

Simplify the Expression

Subtracting the exponents gives us \(y^{4}\).
04

Result

The simplified form of \( \frac{y^{7}}{y^{3}} \) is \(y^{4}\).

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Key Concepts

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

Rule of Exponents
Understanding the rule of exponents is key to simplifying expressions such as \( \frac{y^{7}}{y^{3}} \). When dealing with exponents, one fundamental rule is that when you divide two numbers with the same base, you subtract the exponents. This rule is universal for all exponential expressions that share the same base.
  • For example, in \( \frac{y^a}{y^b} \), the base is 'y', and the expression simplifies to \( y^{a-b} \).
  • This serves as a shortcut to simplifying many expressions, saving time and reducing complexity in calculations.
Always remember: while the bases must be identical to apply this rule, the numbers used as exponents can vary as needed. Practicing this will make algebraic operations much easier!
Division of Exponents
In the exercise \( \frac{y^{7}}{y^{3}} \), the division of exponents helps simplify the expression. When dividing exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
  • Start by identifying both the exponents: 7 (numerator) and 3 (denominator).
  • Then, perform the subtraction: \( 7 - 3 \) to get \( 4 \).
  • The expression now reorganizes to \( y^4 \).
This makes it clear how the division of exponents turns a potentially confusing fraction into a simplified exponent form like \( y^4 \). This approach is practical or crucial for simplification tasks in algebra.
Algebraic Simplification
Algebraic simplification involves reducing an expression to its simplest form, making it easier to understand and work with. By using the rule of exponents and division of exponents, we effectively simplify expressions and solve algebra problems more efficiently.
  • Firstly, identify any opportunities to apply basic rules, like the rule of exponents, to rewrite parts of the expression more simply.
  • Simplifying like terms can significantly reduce the complexity of expressions.
  • Expressions like \( \frac{y^{7}}{y^{3}} \) become \( y^4 \) after simplification.
The ultimate goal is to make expressions easier to read and use, aiding in solving problems and understanding algebra better. This process requires both the recognition of patterns and the application of fundamental algebraic principles.

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Most popular questions from this chapter

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