Problem 62 - College Algebra with Modeling and Visualization

Question

Exercises \(55-64:\) Use the power rules to simplify the expression. Use positive exponents to write your answer. $$ \left(\frac{-3}{x^{3}}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{9}{x^6} \).

Step 1: Apply the Power of a Quotient Rule

The power of a quotient rule states that you can distribute the power to both the numerator and the denominator. So, apply the square to both \( -3 \, ext{and} \, x^3 \, ext{separately}.\) Thus, \[ \left( \frac{-3}{x^3} \right)^2 = \frac{(-3)^2}{(x^3)^2} \]

Step 2: Simplify the Numerator

Simplify \( (-3)^2 \) by multiplying \( -3 \) by itself.So, \( (-3)^2 = 9 \).

Step 3: Simplify the Denominator

Apply the power rule \( (a^m)^n = a^{m \cdot n} \) to the denominator. Calculate \( (x^3)^2 = x^{3 \cdot 2} = x^6 \).

Step 4: Write the Final Expression with Positive Exponents

Write the expression with positive exponents as ensured by previous steps:Thus, \\[ \frac{9}{x^6} \]

Key Concepts

Power Rule

In algebra, the Power Rule is a shortcut that helps simplify expressions involving exponents. It is particularly useful when dealing with powers raised to another power. The rule states that \((a^m)^n = a^{m \cdot n}\), which means that when you raise a power to another power, you multiply the exponents.

In the exercise, we used the Power Rule on the denominator \( (x^3)^2 \). Here's how it works:

• Identify the base \( x \) and the exponents \( m=3 \) and \( n=2 \).
• Multiply the exponents together: \( 3 \cdot 2 = 6 \).
• The expression simplifies to \( x^6 \).

Applying this rule can make complex expressions much simpler and easier to handle. It is a foundational tool in algebra when working with polynomials and other expressions involving powers.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to a specific power, called the exponent. This operation tells us how many times to multiply the base by itself. For instance, \((-3)^2 = (-3) \times (-3)\), resulting in 9.

Here is how the concept of exponentiation applies to different parts of the process:

• To simplify \((-3)^2\), recognize that the base is \(-3\) and the exponent is 2.
• Multiply the base by itself as many times as indicated by the exponent.
• This results in \( 9 \) because multiplying two negative numbers results in a positive product.

Understanding exponentiation is crucial, especially when negative numbers or fractions are involved, as it aids in achieving the correct sign and value in the final expression.

Simplifying Expressions

Simplifying expressions in algebra involves breaking down complex fractions or expressions to their simplest form, often by using algebraic rules like the Power Rule or operations like exponentiation. The aim is to make the expression as straightforward as possible while maintaining its value.

In our original example, simplifying the expression \[ \left( \frac{-3}{x^3} \right)^2 \]involved several steps:

• Applying the Power Rule to each part of the fraction to separate the numerator and denominator.
• Simplifying the numerator by calculating \((-3)^2\) to achieve the positive result \(9\).
• Using exponentiation to handle the denominator, \(x^3\) raised to the power 2, resulting in \(x^6\).

The end result was the simplified expression \[ \frac{9}{x^6} \],which only involves positive exponents, aligning with the problem's requirements. Mastering the skill of simplifying expressions can save time and reduce errors, making it an essential skill in algebra.