BLAST FROM THE PAST Simplify each expression. 1.) \( w^{4} \cdot w^{6}= \) \( \qquad \) 24 2.) \( \left(8^{5}\right)^{3}= \) \( \qquad \) \( \lambda \) 3.) \( n^{3} \cdot n^{4}= \) \( \qquad \) 0.75 4.) \( 2 m^{6} \cdot 5 m^{3}= \) \( \qquad \) \( 10^{2} \) 5.) \( \frac{x^{4}}{x^{4}}= \) \( \qquad \) \( \frac{16}{6} \) 6.) \( \left[\left(7^{3}\right)^{2}\right]^{5}= \) \( \qquad \) 3 7.) \( \frac{4^{8}}{4^{6}}= \) \( \qquad \) \( \frac{2.5}{1.5} \) 8.) \( 6^{12} \cdot 6^{3}=36 \)

1.) \( w^{4} \cdot w^{6} \)   Using the law of exponents for multiplication, add the exponents:   \( w^{4} \cdot w^{6} = w^{4+6} = w^{10} \). 2.) \( \left(8^{5}\right)^{3} \)   Raise a power to a power by multiplying exponents:   \( \left(8^{5}\right)^{3} = 8^{5\cdot3} = 8^{15} \).   (Alternatively, express \(8\) as \(2^3\) to get:   \( 8^{15} = \left(2^{3}\right)^{15} = 2^{45} \).) 3.) \( n^{3} \cdot n^{4} \)   Add the exponents when multiplying like bases:   \( n^{3} \cdot n^{4} = n^{3+4} = n^{7} \). 4.) \( 2 m^{6} \cdot 5 m^{3} \)   Group the numerical coefficients and the powers of \(m\):   \( 2 \cdot 5 = 10 \) and \( m^{6} \cdot m^{3} = m^{6+3} = m^{9} \).   Thus, \( 2 m^{6} \cdot 5 m^{3} = 10 m^{9} \). 5.) \( \frac{x^{4}}{x^{4}} \)   Subtract the exponents in the quotient (or note that any nonzero number divided by itself equals 1):   \( \frac{x^{4}}{x^{4}} = x^{4-4} = x^{0} = 1 \) (for \( x\neq0 \)). 6.) \( \left[\left(7^{3}\right)^{2}\right]^{5} \)   Multiply the exponents sequentially:   First, \( \left(7^{3}\right)^{2} = 7^{3\cdot2} = 7^{6} \).   Then, \( \left(7^{6}\right)^{5} = 7^{6\cdot5} = 7^{30} \). 7.) \( \frac{4^{8}}{4^{6}} \)   Subtract the exponents when dividing like bases:   \( \frac{4^{8}}{4^{6}} = 4^{8-6} = 4^{2} = 16 \). 8.) \( 6^{12} \cdot 6^{3} \)   Add the exponents when multiplying like bases:   \( 6^{12} \cdot 6^{3} = 6^{12+3} = 6^{15} \).