Simplify ((c^-2)^3d^4)/(d^-2)

Solution:

Step 1

Apply the negative exponent rule to bring $d^{-2}$ to the numerator: $\left(c^{-2}\right)^3 \cdot d^4 \cdot d^2$.

Step 2

Combine like bases by adding their exponents.

Step 2.1

Position $d^2$ next to $d^4$: $\left(c^{-2}\right)^3 \cdot d^2 \cdot d^4$.

Step 2.2

Utilize the exponent addition rule $a^m \cdot a^n = a^{m+n}$: $\left(c^{-2}\right)^3 \cdot d^{2+4}$.

Step 2.3

Complete the addition of exponents: $\left(c^{-2}\right)^3 \cdot d^6$.

Step 3

Handle the exponentiation of an exponent by multiplying the exponents.

Step 3.1

Apply the rule $\left(a^m\right)^n = a^{mn}$: $c^{-2 \cdot 3} \cdot d^6$.

Step 3.2

Perform the multiplication of exponents: $c^{-6} \cdot d^6$.

Step 4

Convert the negative exponent to a fraction using the rule $b^{-n} = \frac{1}{b^n}$: $\frac{1}{c^6} \cdot d^6$.

Step 5

Combine the fraction and the remaining term: $\frac{d^6}{c^6}$.

Knowledge Notes:

The problem-solving process involves simplifying an algebraic expression using exponent rules. The relevant knowledge points include:

1. Negative exponent rule: $b^{-n} = \frac{1}{b^n}$, which allows us to transform a term with a negative exponent into a fraction.

2. Power of a power rule: $\left(a^m\right)^n = a^{mn}$, which is used when an exponentiated term is raised to another power.

3. Product of powers rule: $a^m \cdot a^n = a^{m+n}$, which is applied when multiplying terms with the same base.

4. Simplification of algebraic expressions: The process of reducing expressions to their simplest form by applying arithmetic operations and algebraic rules.

These rules are essential for manipulating and simplifying expressions involving exponents, which is a fundamental skill in algebra.