Problem 117 - Elementary Algebra

Question

Simplify the following problems. $$ \left(\frac{3 x^{2}}{4 y^{3}}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $$\left(\frac{3 x^{2}}{4 y^{3}}\right)^{2}$$ Answer: $$\frac{9x^{4}}{16y^{6}}$$

Step 1: Rewrite the expression

Rewrite the expression without the parenthesis, as follows: $$ \left(\frac{3 x^{2}}{4 y^{3}}\right)^{2} = \frac{3^2 x^{2 \cdot 2}}{4^2 y^{3 \cdot 2}} $$

Step 2: Apply exponent rules

Using the rules of exponentiation, distribute the exponent (2) to both the numerator and the denominator of the fraction separately: $$ \frac{3^2 x^{2 \cdot 2}}{4^2 y^{3 \cdot 2}} = \frac{9x^{4}}{16y^{6}} $$

Step 3: Simplify the expression

The simplified expression is the result obtained in Step 2: $$ \frac{9x^{4}}{16y^{6}} $$

Key Concepts

Simplifying Expressions

Expression simplification is about reducing an expression to its simplest or most easily understandable form. This often involves following basic algebraic rules and operations to make expressions more manageable. In this exercise, the process starts with an expression involving parentheses and exponents. Here are the key steps when simplifying such expressions:

• Focus on eliminating parentheses: Distribute powers across the term inside the parentheses.
• Simplify within the expression: Combine like terms and use arithmetic operations when necessary.
• Consider multi-step simplification: Sometimes, you may need to simplify in stages. Each stage should bring you closer to the simplest form.

By carefully applying these strategies and rules, like breaking down components of the expression and working through them systematically, you simplify complex expressions into their base parts.

Algebraic Fractions

Algebraic fractions are like regular numerical fractions, but they include variables. They follow the same basic principles as numerical fractions but with added layers of complexity due to these variables.When simplifying algebraic fractions:

• Apply the same rules for fractions: Look for common factors in the numerator and denominator that you can simplify.
• Bring variables and coefficients under control: Use these rules alongside those for handling variables.
• Separate complex operations: Break down the problem into smaller parts to make it manageable.

Consider the fraction \( \frac{3 x^{2}}{4 y^{3}} \), where each term inside the fraction can be simplified or operated on separately. When raising to a power, distribute the power to both the numerator and the denominator.

Exponentiation

Exponentiation is a crucial concept in algebra, especially when dealing with simplifying expressions. Here, it refers to raising numbers or variables to a power, which is a way of denoting repeated multiplication.Key points about exponentiation:

• Applying powers: When you apply a power to a fraction, the power must be distributed to both the numerator and the denominator.
• Multiplying exponents: If you have a power raised to another power, you multiply the exponents (e.g., \( (x^a)^b = x^{a \cdot b} \)).
• Simplifying results: As seen in the exercise, apply these multiplication rules and simplify the resulting expression.

In our existing problem, exponentiation transforms \( \left(\frac{3 x^{2}}{4 y^{3}}\right)^{2} \) into a simpler form by calculating \( 3^2 = 9 \) and \( 4^2 = 16 \), and multiplying exponents for the variables, producing \( x^4 \) and \( y^6 \). This understanding of exponentiation helps simplify more complex algebraic fractions by breaking them down according to these rules.