Problem 124 Simplify. Assume no division by ... [FREE SOLUTION]

Chapter 4: Problem 124

Simplify. Assume no division by 0. $$\left(\frac{y^{3} y}{2 y y^{2}}\right)^{3}$$

Short Answer

Expert verified

The simplified form of $\left(\frac{y^{3} y}{2 y y^{2}}\right)^{3}$ is $\frac{y^3}{8}$.

Step by step solution

Simplify Inside the Brackets

The first step is to simplify inside the brackets. Here, the expression inside brackets is $\frac{y^{3} y}{2 y y^{2}}$. First simplify the top of the fraction by adding the exponents of $y$ as they are multiplied together, $y^{3} y$ becomes $y^{(3+1)}$ or $y^4$. Similarly simplify the bottom of the fraction $2 y y^{2}$ becomes $2y^{(1+2)}$ or $2y^3$. So, the expression inside brackets is $\frac{y^4}{2y^3}$.

Perform Division

The next step is to perform the division where needed. Here, the expression has become $\frac{y^4}{2y^3}$. Dividing $y^4$ by $y^3$ will result in $y^{(4-3)}$ or $y$. Therefore, after division, the simplified expression inside brackets is $\frac{y}{2}$.

Apply Exponent to the Simplified Expression

Finally, apply exponent 3 to the simplified expression, $\left(\frac{y}{2}\right)^3$. This means $y$ is cubed and 2 is cubed, results in $y^3$ and 8. Therefore, the final simplified expression is $\frac{y^3}{8}$.

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Key Concepts

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

Exponent Rules

Exponents are a mathematical shorthand to express repeated multiplication. They are powerful tools in algebra, especially when simplifying complex expressions. Let's consider some essential exponent rules that will make simplification easier:

Product Rule: When you multiply two powers with the same base, add their exponents. For example, for bases like $y^m$ and $y^n$, the product rule is $y^m \times y^n = y^{(m+n)}$.
Power Rule: When you raise an exponent to another exponent, multiply the exponents together. That is, $(y^m)^n = y^{(m \times n)}$.
Quotient Rule: When dividing two powers with the same base, subtract the exponents. Hence, $\frac{y^m}{y^n} = y^{(m-n)}$.

These rules enable you to break down and simplify expressions quickly. In our original exercise, we applied the product rule to combine like terms initially, followed by the quotient rule to simplify. Understanding and applying these rules allows for a clear and straightforward path to simplification.

Fraction Simplification

Fraction simplification is about reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. Simplifying fractions involves these key elements:

Combining Like Terms: In the initial simplification, it's crucial to combine like terms, especially when dealing with variables. This means adding or subtracting exponents when terms share the same base.
Identifying Common Factors: Look for common factors in the numerator and denominator. This helps in canceling out terms to simplify the fraction further. In our example, we had $\frac{y^4}{2y^3}$. Here $y^3$ appears in both terms, allowing for simplification.
Reducing to Lowest Terms: This happens once like terms are combined, and all common factors between the numerator and the denominator have been eliminated.

In our exercise, simplifying $\frac{y^4}{2y^3}$ to $\frac{y}{2}$ helped us see that division has a significant role in simplifying fractions when algebra is involved. By focusing on these elements, you can make complex fractions much simpler to solve.

Polynomial Division

Polynomial division is a method used to simplify expressions that can be intimidating at first. Let's demystify it by breaking down the process:

Understanding Division of Terms: With polynomial division, each term of the polynomial is divided separately. This technique is straightforward and closely tied to fraction simplification.
Applying Exponent Rules: As you divide polynomials, use exponent rules like the quotient rule to manage the variable terms, making the expressions simpler. For instance, simplifying $\frac{y^4}{y^3}$ results in $y^{4-3} = y$.
Resulting Terms: Once division is performed, collect the resulting terms to form a simpler polynomial expression or fraction. In our task, the result was $\frac{y}{2}$, a fraction involving both constants and variables.

By mastering polynomial division, algebraic simplification becomes less daunting. It offers a systematic way to handle more complex expressions by breaking them down into smaller, more manageable components.

91影视

Simplify. Assume no division by 0. $$\left(\frac{y^{3} y}{2 y y^{2}}\right)^{3}$$

Short Answer

Step by step solution

Simplify Inside the Brackets

Perform Division

Apply Exponent to the Simplified Expression

Key Concepts

Exponent Rules

Fraction Simplification

Polynomial Division

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