Problem 25 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(6 x^{-3} y^{5}\right)\left(-7 x^{2} y^{-9}\right)$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(-42x^{-1}y^{-4}\).

Step 1: Multiply constant terms

Multiply the constant terms 6 and -7: \[ (6)\times(-7) = -42 \]

Step 2: Multiply variables with the same base

Add the exponents of x (x^{-3} and x^{2}) and y (y^{5} and y^{-9}): For x: \[ (-3) + 2 = -1 \] For y: \[ 5 + (-9) = -4 \]

Step 3: Rewrite the expression with simplified terms

Combine the results from steps 1 and 2 to write the simplified expression: \[ -42x^{-1}y^{-4} \] The simplified form of the given expression is \[ -42x^{-1}y^{-4} \]

Key Concepts

Simplifying Expressions

Simplifying expressions in mathematics is a process where you rewrite an expression in a more compact and comprehensible form. This often involves combining like terms, reducing fractions, or factoring out common elements. When dealing with expressions that involve exponents, simplification not only makes your equations more manageable but also clearer to understand.

Take, for example, the expression given in the exercise: \( (6x^{-3}y^{5})(-7x^{2}y^{-9}) \). The simplification process follows a logical series of operations whereby:

• You start by dealing with the constants (6 and -7) and variables separately.
• Next, you apply rules specific to exponents.
• Finally, you rewrite the expression in its simplified form.

In this example, the product of constant terms \(6\) and \(-7\) results in \(-42\). This value becomes part of the simplified expression. Understanding these basic steps of simplification is crucial to solving more complex algebraic problems effectively.

Negative Exponents

Exponents are used to denote repeated multiplication of a base number. But what happens when those exponents are negative? Negative exponents represent the reciprocal of the base with a positive exponent. For any non-zero number \(a\), the expression \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\). Here's how this connects to the original problem:

In the expression \( -42x^{-1}y^{-4} \), you encounter negative exponents \(x^{-1}\) and \(y^{-4}\).

• The term \(x^{-1}\) can be rewritten as \(\frac{1}{x^1} = \frac{1}{x}\).
• Similarly, \(y^{-4}\) becomes \(\frac{1}{y^4}\).

This interpretation is key to understanding how to rewrite expressions and potentially calculate specific values. Knowing that negative exponents flip the base in a fraction helps you not only simplify your results but also transition smoothly between mathematical forms.

Multiplication of Variables

Multiplying variables involves not just understanding basic multiplication, but also applying the unique rules of exponents. When you're multiplying expressions that have the same base and different exponents, you simply add the exponents together. This principle is considered the fundamental rule of multiplying powers..

Using this rule in our exercise:

• The base \(x\) is found in both terms, with exponents \(-3\) and \(2\). You add these: \((-3) + 2 = -1\).
• The base \(y\) has exponents \(5\) and \(-9\): \(5 + (-9) = -4\).

The key outcome here is that when multiplying two terms with the same bases, you become adept at combining the exponents correctly. This rule creates a foundational understanding necessary for mastering algebra and other higher-level math topics.