Math worksheet matching exponent rules to equations.

Problem Analysis:

The task involves matching each equation with the correct law of exponents used in simplifying it. Additionally, there is a riddle to solve based on the context provided.

#### Riddle:
"We work in the same suit – usually white, fixing living apparatuses. Who am I?"

This riddle describes medical professionals who wear white suits and work in healthcare settings. The answer to the riddle is "doctor" or "nurse", but since the context mentions "fixing living apparatuses," the most fitting answer is "doctor."

#### Equations and Laws of Exponents:
We need to match each equation with the appropriate law of exponents:

1. \((-3a^3)^2 = 9a^6\)
2. \((a^3b^0)^5 = a^{15}\)
3. \((a^5b^2)(a^2b^4) = a^7b^6\)
4. \(\frac{32a^5}{8a^2} = 4a^3\)
5. \((12a^2b^5)^0 = 1\)
6. \(\frac{a^{-2}}{b^{-3}} = \frac{b^3}{a^2}\)

The laws of exponents provided are:
- R: Negative Exponent
- O: Zero Exponent
- D: Power Rule
- T: Quotient Rule
- O: Power Rule and Zero Exponent
- C: Product Rule

Step-by-Step Solution:

#### Equation 1: \((-3a^3)^2 = 9a^6\)
- Here, we are raising a product to a power. The Power Rule states that \((xy)^n = x^n y^n\).
- Applying the Power Rule:
\[
(-3a^3)^2 = (-3)^2 \cdot (a^3)^2 = 9 \cdot a^{3 \cdot 2} = 9a^6
\]
- Match: D (Power Rule)

#### Equation 2: \((a^3b^0)^5 = a^{15}\)
- Here, we use the Power Rule and the Zero Exponent Rule. The Zero Exponent Rule states that \(x^0 = 1\) for any \(x \neq 0\).
- Applying the rules:
\[
(a^3b^0)^5 = (a^3 \cdot 1)^5 = (a^3)^5 = a^{3 \cdot 5} = a^{15}
\]
- Match: O (Power Rule and Zero Exponent)

#### Equation 3: \((a^5b^2)(a^2b^4) = a^7b^6\)
- Here, we are multiplying powers with the same base. The Product Rule states that \(x^m \cdot x^n = x^{m+n}\).
- Applying the Product Rule:
\[
(a^5b^2)(a^2b^4) = a^{5+2} \cdot b^{2+4} = a^7b^6
\]
- Match: C (Product Rule)

#### Equation 4: \(\frac{32a^5}{8a^2} = 4a^3\)
- Here, we are dividing powers with the same base. The Quotient Rule states that \(\frac{x^m}{x^n} = x^{m-n}\).
- Applying the Quotient Rule:
\[
\frac{32a^5}{8a^2} = \frac{32}{8} \cdot \frac{a^5}{a^2} = 4 \cdot a^{5-2} = 4a^3
\]
- Match: T (Quotient Rule)

#### Equation 5: \((12a^2b^5)^0 = 1\)
- Here, we use the Zero Exponent Rule, which states that any non-zero number raised to the power of 0 is 1.
- Applying the Zero Exponent Rule:
\[
(12a^2b^5)^0 = 1
\]
- Match: O (Zero Exponent)

#### Equation 6: \(\frac{a^{-2}}{b^{-3}} = \frac{b^3}{a^2}\)
- Here, we use the Negative Exponent Rule, which states that \(x^{-n} = \frac{1}{x^n}\).
- Applying the Negative Exponent Rule:
\[
\frac{a^{-2}}{b^{-3}} = \frac{\frac{1}{a^2}}{\frac{1}{b^3}} = \frac{b^3}{a^2}
\]
- Match: R (Negative Exponent)

Final Matches:

1. \((-3a^3)^2 = 9a^6\) → D (Power Rule)
2. \((a^3b^0)^5 = a^{15}\) → O (Power Rule and Zero Exponent)
3. \((a^5b^2)(a^2b^4) = a^7b^6\) → C (Product Rule)
4. \(\frac{32a^5}{8a^2} = 4a^3\) → T (Quotient Rule)
5. \((12a^2b^5)^0 = 1\) → O (Zero Exponent)
6. \(\frac{a^{-2}}{b^{-3}} = \frac{b^3}{a^2}\) → R (Negative Exponent)

Answer to the Riddle:

The riddle describes a medical professional who works in a white suit and fixes living apparatuses. The answer is "doctor."

Final Answer:

\[
\boxed{\text{doctor}}
\]