Problem 43 - Precalculus Mathematics for Calculus

Question

Simplify the expression and eliminate any negative exponents(s). (a) \(\left(5 x^{2} y^{3}\right)\left(3 x^{2} y^{5}\right)^{4}\) (b) \(\left(2 a^{3} b^{2}\right)^{2}\left(5 a^{2} b^{5}\right)^{3}\)

Step-by-Step Solution

Answer

(a) Simplifies to \(405 x^{10} y^{23}\); (b) Simplifies to \(500 a^{12} b^{19}\).

Step 1: Expand Expressions

Let's expand the expression for part (a): \[ \left(5 x^{2} y^{3}\right)\left(3 x^{2} y^{5}\right)^{4} \]Start by expanding the second term: \[ (3 x^{2} y^{5})^{4} = 3^{4} \cdot (x^{2})^{4} \cdot (y^{5})^{4} \].This becomes:\[ 81 \cdot x^{8} \cdot y^{20} \].

Step 2: Combine Expressions

Continue simplifying part (a) by substituting the expanded expression back:\[ 5 x^{2} y^{3} \cdot 81 x^{8} y^{20} \].Combine like terms:\[ 5 \cdot 81 \cdot x^{2+8} \cdot y^{3+20} \].This simplifies to:\[ 405 x^{10} y^{23} \].

Step 3: Expand Using Power Laws

For part (b), expand each term separately:\[ (2 a^{3} b^{2})^{2} = 2^{2} (a^{3})^{2} (b^{2})^{2} = 4 a^{6} b^{4} \] and \[ (5 a^{2} b^{5})^{3} = 5^{3} (a^{2})^{3} (b^{5})^{3} = 125 a^{6} b^{15} \].

Step 4: Combine Expanded Expressions

Multiply the expanded terms in part (b):\[ 4 a^{6} b^{4} \cdot 125 a^{6} b^{15} \].Combine like terms:\[ 4 \cdot 125 \cdot a^{6+6} \cdot b^{4+15} \].This simplifies to:\[ 500 a^{12} b^{19} \].

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing an expression to its most efficient and understandable form. Let's take an expression involving exponents:
\( \left(5 x^{2} y^{3}\right)\left(3 x^{2} y^{5}\right)^{4} \). The goal is to simplify this expression so that it is easier to understand and work with.
To simplify, we first look for opportunities to apply mathematical rules such as exponent rules and arithmetic procedures. In our example, the expression \( \left(3 x^{2} y^{5}\right)^{4} \) can be expanded using the power law. Simply raising each factor inside to the power of four results in multiplying the exponents, as seen in \( 3^{4} \cdot (x^{2})^{4} \cdot (y^{5})^{4} \).
Perform calculations where necessary to reach the final expression. Reducing complex expressions makes future calculations less cumbersome and increases clarity. Simplified expressions help you quickly recognize potential errors and improve overall problem-solving efficiency.

Power Laws

Power laws, or the laws of exponents, are crucial when working with expressions containing exponents. These rules help simplify expressions and solve equations more efficiently.
Here are some key power laws to remember:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
• Power of a Product: \( (ab)^n = a^n \cdot b^n \)

In the provided exercise, power laws were employed to expand and simplify expressions. For instance, when given \( (2 a^{3} b^{2})^{2} \), each term in the parentheses is individually raised to the second power, resulting in \( 2^{2} \cdot (a^{3})^{2} \cdot (b^{2})^{2} \).
Understanding and using power laws allows you to efficiently manage and manipulate expressions, setting a solid foundation for tackling more challenging problems.

Combining Like Terms

Combining like terms is an essential step in simplifying expressions and solving equations. Like terms are terms in an expression that have identical variable parts; they can be combined by adding or subtracting their coefficients.
In the expressions \( 5 x^{2} y^{3} \cdot 81 x^{8} y^{20} \) and \( 4 a^{6} b^{4} \cdot 125 a^{6} b^{15} \), combining like terms involves bringing together the coefficients while adding the exponents of the same variables. For example:

• Combine \( x^{2} \) and \( x^{8} \) by adding exponents: \( x^{2+8} = x^{10} \)
• Combine \( y^{3} \) and \( y^{20} \) by adding exponents: \( y^{3+20} = y^{23} \)
• Multiply coefficients: \( 5 \cdot 81 = 405 \)

Combining like terms simplifies expressions and makes them clearer and easier to work with, minimizing potential mistakes and enhancing algebraic skills.