Exponents

1. Which operation is the opposite of multiplication?

   Solution

   Division

---

2. In the expression, \( (14x^2) \) what is the coefficient?

   Solution

   The coefficient of $14x^2 \textit{ is } 14$.

---

3. In the expression, $ (x^3) $ what is the coefficient?

   Hint

   What hidden value must exist for this expression to equal more than zero?

   Solution

   The coefficient of $ (x^3) $ is 1.

---

4. Solve: \( (8m^2)(3m^3) \)

   Solution

   \begin{align} (8m^2)(3m^3) &= (8 \times 3)(m^2m^3) \\ &= 24m^5 \end{align}

---

5. \( (-3a^2b^4)(-2b) \)

   Solution

   \begin{align} (-3 a^2 b^4)(-2b) &= (-3 \times -2)(a^2 b^4 b^1) \\ &= 6 a^2 b^5 \end{align}

---

6. Solve: \( \dfrac{-24 a^{10} b^3}{5 a^3} \)

   Solution

   \begin{align} \dfrac{-24 a^{10} b^3}{5 a^3} &= \left ( \dfrac{-25}{5} \right ) \left ( \dfrac{a^{10}}{a^3} \right ) \left ( \dfrac{b^3}{b^0} \right ) \\ &= -5 a^7 b^3 \end{align}

---

7. Solve: \( (-2xy^2)^6 \)

   Solution

   \begin{align} (-2xy^2)^6 &= (-2^6)(x^6)(y^{2\times6}) \\ &= 64x^6y^{12} \end{align}

---

8. Simplify: \( 36^{\frac{{\color{red}1}}{{\color{teal}2}}} \)

   Solution

   \begin{align} 36^{\frac{{\color{red}1}}{{\color{teal}2}}} &= \sqrt[{\color{teal}2}]{36}^{{\color{red},1}} \\ &= \sqrt[{\color{teal}2}]{{\color{teal}(6)(6)}}^{{\color{red},1}} \\ &= 6^{{\color{red}1}} \\ &= 6 \end{align}

---

9. Simplify: \( 144^{\frac{{\color{red}1}}{{\color{teal}2}}} \)

   Solution

   \begin{align} 144^{\frac{{\color{red}1}}{{\color{teal}2}}} &= \sqrt[{\color{teal}2}]{144}^{{\color{red},1}} \\ &=\sqrt[{\color{teal}2}]{{\color{teal}(12)(12)}}^{{\color{red},1}} \\ &= 12^{{\color{red}1}} \\ &= 12 \end{align}

---

10. Simplify: \( 81^{\frac{{\color{red}3}}{{\color{teal}4}}} \)

    Solution

    \begin{align} 81^{\frac{{\color{red}3}}{{\color{teal}4}}} &= \sqrt[{\color{teal}4}]{81}^{{\color{red},3}} \\ &= \sqrt[{\color{teal}4}]{{\color{teal}(3)(3)(3)(3)}}^{{\color{red},3}} \\ &= 3^{{\color{red}3}} \\ &= 27 \end{align}

---