Problem 47 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \left(-9 x^{3} y\right)\left(-2 x^{6} y^{4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(18x^{9}y^{5}\)

Step 1: Understand the expression

The given expression is \(-9x^{3}y\)*(-2x^{6}y^{4}\). This is the product of two terms. We also note that the bases of the exponents are the same, which is \(x\) and \(y\). Therefore, we can apply the properties of exponents to simplify this expression.

Step 2: Apply properties of exponents

The property of exponential expressions we use here is: \(a^{m}\)*\(a^{n}\)= \(a^{m+n}\), where \(a\) is not equal to zero. Also, we multiply the coefficients of each term.From the given expression, -9 and -2 are the coefficients, therefore we multiply these to get 18. Applying this rule to the expression gives, \(18x^{3+6}y^{1+4}\).

Step 3: Simplify

To simplify further, sum up the exponents and resolve the multiplication. The resulting simplified expression will be \(18x^{9}y^{5}\)

Key Concepts

Properties of Exponents

When working with exponential expressions, it's crucial to understand the properties that allow us to simplify them effectively. One of the fundamental properties of exponents is the Product of Powers property. This states that when you multiply two exponents with the same base, you can add the exponents together. For example, if you have \(a^m\) and \(a^n\), the simplified form is \(a^{m+n}\). This property makes it easier to deal with expressions that have large exponents, by combining them into a single power.
In the original exercise, the expression \((-9x^3y)\times(-2x^6y^4)\) involved the bases \(x\) and \(y\), where each base was raised to different powers. By grouping like bases and using the property \(a^m\times a^n = a^{m+n}\), the exponents of \(x\) (which are 3 and 6) and \(y\) (which are 1 and 4) were added together to create a simplified expression. This is a powerful technique for managing and simplifying expressions involving exponents.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form by performing arithmetic operations and applying mathematical rules. In the context of exponential expressions, this typically involves applying the properties of exponents and combining like terms. It's essential to treat each component separately but consistently within an expression.
In the expression \((-9x^3y)\times(-2x^6y^4)\), simplifying began with the coefficients, which were \(-9\) and \(-2\). Multiplying these coefficients resulted in 18, as shown in the solution. Next, the exponents were combined using the Product of Powers rule, resulting in the expression \(18x^{3+6}y^{1+4}\).
The final step in simplifying was to perform any necessary arithmetic involving the exponents, which yielded the expression \(18x^9y^5\). This concise form makes it much easier to understand and use in further calculations or applications, free of unnecessary complexity.

Exponents

Exponents are a shorthand way to indicate that a number, known as the base, is multiplied by itself a specific number of times. For example, \(x^3\) indicates that the base \(x\) is multiplied by itself three times, or \(x \times x \times x\).
Exponential notation provides a convenient way to represent large numbers or repetitive multiplications, enhancing clarity and efficiency in mathematical expressions. Various rules and properties help in manipulating and simplifying them effectively, such as the Product of Powers property we discussed.
Understanding how to work with exponents is essential for simplifying expressions, solving equations, and modeling real-world phenomena. In the exercise you've seen, working with exponents of like bases allowed the expression \((-9x^3y)\times(-2x^6y^4)\) to be simplified step by step. The outcome was a much clearer expression, \(18x^9y^5\), that retains all the original mathematical information in a more manageable form.