Problem 51 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(5 x^{2} y^{3}\right)\left(3 x^{2} y^{5}\right)^{4} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 405 x^{10} y^{23} \).

Step 1: Distribute the exponent

Apply the exponent of 4 to each element in the expression \( \left(3x^2y^5\right)^4 \). This transforms the expression into \( 3^4 x^{2 \cdot 4} y^{5 \cdot 4} = 81 x^8 y^{20} \).

Step 2: Multiply the expressions

Multiply the simplified expression from Step 1 by the initial expression: \( (5x^2y^3)(81x^8y^{20}) \).

Step 3: Multiply the coefficients

Multiply the coefficients together: \(5 \times 81 = 405 \). This gives us \( 405 \times x^2 y^3 \times x^8 y^{20} \).

Step 4: Add the exponents

Since the bases are the same, add the exponents for corresponding variables: for \(x\), add the exponents: \(2+8=10\); and for \(y\), add the exponents: \(3+20=23\). This results in \( 405 x^{10} y^{23} \).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent tells us how many times to multiply the base by itself. In other words, it's a shorthand way of expressing repeated multiplication. For example, the expression \(x^3\) means \(x \times x \times x\).

In the problem, we see \((3x^2y^5)^4\). Here, \(3x^2y^5\) is the base, and 4 is the exponent. We apply the exponent to distribute it to each factor in the base, resulting in \(3^4\), \(x^{2 \times 4}\), and \(y^{5 \times 4}\). Each component of the base is raised to the fourth power independently.

Distributive Property

The distributive property is a helpful algebraic rule that allows us to multiply a single term by every term inside a parenthesis. It's commonly expressed as \(a(b + c) = ab + ac\).

In our scenario, we are effectively utilizing the distributive property when we apply the exponent to the terms inside \((3x^2y^5)^4\). Instead of directly adding or multiplying, we "distribute" the exponent to each part of the expression, transforming it into \(3^4 \times x^{2 \times 4} \times y^{5 \times 4}\). This ensures that each element is equally affected by the exponentiation.

Coefficient Multiplication

Coefficient multiplication involves multiplying the numerical parts of algebraic terms while keeping the variable parts separate for the moment.

For the given problem, we multiply the coefficients of the terms \((5x^2y^3)\) and \((81x^8y^{20})\). The numbers 5 and 81 are the coefficients here. When we multiply them, we get 405, which becomes the new coefficient of our combined expression.

This step focuses on the arithmetic aspect of simplifying the expression, ensuring that the numbers are calculated before dealing with the variables and their exponents.

Exponent Rules

Exponent rules are guidelines that help with handling operations involving powers of numbers or variables.

Some key exponent rules include:

• Product of powers: When multiplying two expressions with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a power: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Power of a product: Apply the exponent to each factor inside the parentheses: \((ab)^n = a^n b^n\).

In the problem, after multiplying coefficients, we apply the product of powers rule for the variables. For \(x\), based both from \(5x^2y^3\) and \(81x^8y^{20}\), we add \(2 + 8\) to get \(x^{10}\). Similarly, for \(y\), we add \(3 + 20\) to obtain \(y^{23}\). These exponent rules simplify calculations and avoid errors while dealing with powers.