Problem 6 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(\frac{c d}{3}\right)^{-2} $$

Step-by-Step Solution

Answer

\( \frac{9}{c^2 d^2} \)

Step 1: Apply the Negative Exponent Rule

To simplify an expression with a negative exponent, take the reciprocal of the base and make the exponent positive. So, replace \( \left( \frac{c d}{3} \right)^{-2} \) with \( \left( \frac{3}{c d} \right)^2 \).

Step 2: Apply the Power of a Fraction Rule

When raising a fraction to a power, raise both the numerator and the denominator to that power. Apply this rule to \( \left( \frac{3}{c d} \right)^2 \), resulting in \( \frac{3^2}{(c d)^2} \).

Step 3: Simplify Exponents

Calculate \( 3^2 \) which equals 9, and express \( (c d)^2 \) as \( c^2 d^2 \). Thus, the expression becomes \( \frac{9}{c^2 d^2} \).

Key Concepts

Fractional Exponents

Fractional exponents are a way to represent roots through the use of powers. In the context of exponents, fractions act as indicators of roots. For instance, if you encounter a term like \( x^{1/2} \), it signifies the square root of \( x \). This relationship can be extended to other types, such as \( x^{1/3} \), representing the cube root.
Understanding fractional exponents can make expressions involving roots simpler to handle and calculate. For example, simplifying \( x^{3/2} \) becomes much easier when you interpret it correctly as the square root of \( x^3 \), or \( (\sqrt{x})^3 \).

• Fractional exponents are a way to express roots.
• The denominator of the fraction represents the root.
• The numerator signifies the power to which the number is raised.

Grasping fractional exponents will enable you to transform roots into powers, thereby simplifying complex expressions.

Power of a Fraction Rule

When it comes to raising a fraction to a power, the power of a fraction rule is a powerful tool. It tells us to apply the exponent to both the numerator and the denominator of the fraction independently. For instance, if you have an expression \( \left( \frac{a}{b} \right)^n \), it can be expressed as \( \frac{a^n}{b^n} \).
This rule is immensely useful when working with more intricate expressions that contain exponents, as it allows you to simplify them effectively and efficiently. Let's break it down with the expression \( \left( \frac{3}{c d} \right)^2 \). Applying the power of a fraction rule, you raise 3 to the power of 2 and simultaneously raise \( c d \) to the power of 2, resulting in \( \frac{3^2}{(c d)^2} \).

• Apply the exponent to the numerator.
• Apply the exponent to the denominator.
• Helps in simplifying complex fraction expressions.

Using this rule consistently allows you to tackle even the most challenging expressions with fractional components.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. This is done by using various mathematical rules and operations to make expressions more manageable for computation or comparison. When working with expressions involving exponents, the fundamental approach is to calculate powers and reorganize terms.
Consider our original expression \( \left( \frac{c d}{3} \right)^{-2} \). After using the negative exponent rule, we swapped the base and made the exponent positive, arriving at \( \left( \frac{3}{c d} \right)^2 \). Applying the power of a fraction rule led us to \( \frac{3^2}{(c d)^2} \), which further simplifies to \( \frac{9}{c^2 d^2} \).

• Apply basic math rules to simplify.
• Transform complex terms to elementary ones.
• Use rules like negative exponent and power of a fraction effectively.

Simplifying expressions helps not just in solving equations but in understanding the inherent relationships within an equation, making them more straightforward to work with.