Problem 62 - Elementary Algebra

Question

Simplify \(\left(\frac{4 a^{2} b}{3 x y^{3}}\right)^{2}\).

Step-by-Step Solution

Answer

Question: Simplify the expression \(((4ab^2)^2)((3x^2y^3)^2)\). Answer: \(\frac{16a^4b^2}{9x^2y^6}\)

Step 1: Apply the power to each term inside the parentheses

Apply the power of 2 to each term inside the parentheses. This means that we need to square each term individually: \((4^2)(a^2)^2(b^2)((3^2)(x^2)(y^3)^2)\).

Step 2: Simplify the powers

Simplify the powers by multiplying the exponents with the power of 2: \((16)(a^4)(b^2)(9x^2y^6)\).

Step 3: Combine terms

Combine the terms in the expression. Since the terms are all multiplied, we just need to multiply the constants and keep the variable terms together: \(\frac{16 \cdot a^4 \cdot b^2}{9 \cdot x^2 \cdot y^6}\).

Step 4: Simplify the expression

The expression is now simplified to its simplest form, which is \(\frac{16a^4b^2}{9x^2y^6}\).

Key Concepts

Simplifying Expressions

When simplifying expressions like \(\left(\frac{4 a^{2} b}{3 x y^{3}}\right)^{2}\), we're working to make them more manageable. Simplification involves reducing the expression to its most basic form without changing its value. Here are a few key points to remember:

• Look for ways to reduce fractions by finding common factors.
• Combine like terms, which involves adding or multiplying coefficients and variables separately.
• Apply mathematical operations such as squaring carefully to each part of the expression.

Take time to follow the order of operations (PEMDAS/BODMAS), which will guide you in processing terms in the correct sequence.

Power Rule

The power rule is vital in simplifying expressions like \(\left(\frac{4 a^{2} b}{3 x y^{3}}\right)^{2}\). This rule states that when you raise a power to another power, you multiply the exponents. For instance, \((a^m)^n = a^{m \cdot n}\). In our expression, we apply the power of 2 to each exponent:

• The coefficient 4 becomes \(4^2 = 16\).
• The variable term \(a^2\) becomes \((a^2)^2 = a^4\).
• The term \(b\) becomes \(b^2\).
• For the denominator, \(3\) becomes \(3^2 = 9\), \(x\) stays the same as \((x)^2\), and \(y^3\) becomes \((y^3)^2 = y^6\).

Each term within the parentheses is squared, following the power rule, resulting in the simplified expression.

Algebraic Fractions

Working with algebraic fractions like \(\frac{16a^4b^2}{9x^2y^6}\) involves understanding both numerical and variable factors. Here's how to simplify them:

• Numerator and Denominator: Separate them and simplify each by factoring and using properties of exponents.
• Combine Terms: Ensure all like terms are combined appropriately. This often involves multiplying coefficients and applying the power rule to like bases.
• Simplify Completely: Always check to make sure there are no more common factors, whether numerical or variable, that can reduce the fraction further.

Algebraic fractions require careful application of both fraction and algebraic rules to reach the simplest form possible.