To raise a monomial to a power, use the POWER RULE: \[ \left(x^{a}\right)^{b}= \] Simplify these monomials. 1. \( \left(x^{2}\right)^{8} \) 2. \( \left(m^{9} n^{5}\right)^{4} \) 3. \( \left(p q^{5}\right)^{3} \) Examples with Coefficients: - Raise the coefficient to the given power. - SIMPLIFY the variables with the power rule. 4. \( \left(5 x^{7}\right)^{2} \) 5. \( \left(2 x^{2} y\right)^{5} \) 6. \( \left(-8 c^{4} d^{9}\right)^{2} \) 7. \( \left(-5 m^{6} n\right)^{4} \) 8. \( \left(-3 x^{2}\right)^{3} \) 9. \( \left(\frac{1}{2} a^{3} b^{4} c^{5}\right)^{3} \)

To simplify these monomials using the power rule, remember that \((x^a)^b = x^{a \cdot b}\). 1. \( \left(x^{2}\right)^{8} = x^{2 \cdot 8} = x^{16} \) 2. \( \left(m^{9} n^{5}\right)^{4} = m^{9 \cdot 4} n^{5 \cdot 4} = m^{36} n^{20} \) 3. \( \left(p q^{5}\right)^{3} = p^{3} q^{5 \cdot 3} = p^{3} q^{15} \) Now, with coefficients: 4. \( \left(5 x^{7}\right)^{2} = 5^{2} (x^{7})^{2} = 25 x^{14} \) 5. \( \left(2 x^{2} y\right)^{5} = 2^{5} (x^{2})^{5} (y)^{5} = 32 x^{10} y^{5} \) 6. \( \left(-8 c^{4} d^{9}\right)^{2} = (-8)^{2} (c^{4})^{2} (d^{9})^{2} = 64 c^{8} d^{18} \) 7. \( \left(-5 m^{6} n\right)^{4} = (-5)^{4} (m^{6})^{4} (n)^{4} = 625 m^{24} n^{4} \) 8. \( \left(-3 x^{2}\right)^{3} = (-3)^{3} (x^{2})^{3} = -27 x^{6} \) 9. \( \left(\frac{1}{2} a^{3} b^{4} c^{5}\right)^{3} = \left(\frac{1}{2}\right)^{3} (a^{3})^{3} (b^{4})^{3} (c^{5})^{3} = \frac{1}{8} a^{9} b^{12} c^{15} \)