Use the Product Rule of Exponents to simplify the expression 75 ⋅72.

Answer 1

We can use the rule $(a^m)\cdot (a^n) = a^{m+n}$ to simplify the expression $75 \cdot 72$. The prime factorization of 75 is $3^1\cdot 5^2$, since $75=3\cdot5\cdot5$, and the prime factorization of 72 is $2^3\cdot 3^2$, since $72=2\cdot2\cdot2\cdot3\cdot3$. Therefore, $$ 75\cdot72=(3^1\cdot 5^2)\cdot(2^3\cdot 3^2).$$ To simplify the expression using the Product Rule of Exponents, we first simplify each factor of $(3^1\cdot 5^2)$ by distributing the exponent over the factors to get $3^1\cdot2^3\cdot 3^2\cdot5^2$. Using the rule $a^m\cdot a^n=a^{m+n}$ again, we can add the exponents in each factor to get \begin{align*}&3^1\cdot2^3\cdot 3^2\cdot5^2\\ &=3^{1+2}\cdot2^3\cdot5^2\\ &=3^3\cdot2^3\cdot5^2\\ &=3^3\cdot2^3\cdot5^2.\end{align*}Using the Product Rule of Exponents one last time, we get the final expression $$3^3\cdot2^3\cdot5^2=3\cdot3\cdot3\cdot2\cdot2\cdot2\cdot5\cdot5=\boxed{27000}.$$