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Chapter 5: Problem 39

\(\left(\frac{3 x^{2}}{4 y^{5}}\right)^{2}\)

Short Answer

Expert verified
\(\frac{9 x^{4}}{16 y^{10}}\)

Step by step solution

01

Understand the problem

Given the expression \(\frac{3 x^{2}}{4 y^{5}}\)^{2}, it needs to be simplified by applying the exponent to both the numerator and the denominator.
02

Apply the exponent to the numerator

Raise the numerator within the parentheses to the power of 2: \((3 x^{2})^{2} = 3^{2} \times (x^{2})^{2} = 9 x^{4}\).
03

Apply the exponent to the denominator

Raise the denominator within the parentheses to the power of 2: \(4 y^{5})^{2} = 4^{2} \times (y^{5})^{2} = 16 y^{10}\).
04

Combine the results

Combine the results from the numerator and denominator: \(\frac{9 x^{4}}{16 y^{10}}\).

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Key Concepts

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

Simplifying Expressions
When simplifying expressions, our goal is to reduce them to their simplest form while preserving their value. This often involves applying arithmetic operations and factoring. In algebra, simplifying helps make complex expressions more manageable.
For example, in the expression \(\frac{3 x^{2}}{4 y^{5}}\)^{2}, applying the exponent to both the numerator and the denominator involves several steps. Simplifying expressions ensures that each term is expressed in its simplest form, which makes it easier to perform subsequent operations or solve equations.
Applying Exponents
Applying exponents is a key part of working with algebraic expressions. An exponent indicates how many times the base number is multiplied by itself.
For instance, in the expression \(\frac{3 x^{2}}{4 y^{5}}\)^{2}, we need to raise both the numerator and the denominator to the power of 2. This means \(3 x^{2}\)^{2} and \(4 y^{5}\)^{2}.
- For the numerator: \((3 x^{2})^{2}\) expands to \(3^{2} \times (x^{2})^{2} = 9 x^{4}\).
- For the denominator: \((4 y^{5})^{2}\) expands to \(4^{2} \times (y^{5})^{2} = 16 y^{10}\).
After applying the exponents, we combine our results to get the simplified expression \(\frac{9 x^{4}}{16 y^{10}}\).
Numerator and Denominator in Fractions
Understanding the numerator and denominator in fractions is crucial for simplifying expressions.
- The numerator is the top part of the fraction and represents the number of parts we have.
- The denominator is the bottom part of the fraction and represents the total number of equal parts the whole is divided into.
In the expression \(\frac{3 x^{2}}{4 y^{5}}\), 3x^{2} is the numerator and 4y^{5} is the denominator.
When we apply the exponent of 2 to the entire fraction, we square both the numerator and denominator separately, giving us \(\frac{9 x^{4}}{16 y^{10}}\).
This step-by-step approach to fractions helps simplify complex algebraic expressions.

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Most popular questions from this chapter

\(\left(x^{2}+13 x+18\right) \div(x+3)\)

\(\left(x^{2}+7 x-18\right) \div(x+9)\)

\(\left(2 x^{2}+7 x+30\right) \div(x+2)\)

\(\left(x^{2}+11 x+15\right) \div(x+3)\)

Problem: Use long division to simplify \(\left(x^{2}-8 x+12\right) \div(x-4)\) Incorrect Answer: $$ \begin{array}{r} x - 4 \longdiv { x ^ { 2 } - 8 x + 1 2 } \\ \frac{-\left(x^{2}-4 x\right)}{-12 x+12} \\ \frac{-(-12 x+48)}{-36} \end{array} $$
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