Simplify negative five 𝑚 raised to the ninth power times negative seven 𝑚 raised to the seventh power.
In this question, we are asked to simplify the product of two algebraic factors. We can see that each factor is a single term and the variable is raised to nonnegative integer exponents. So this is the product of two monomials.
To multiply monomials, we can start by recalling that multiplication is commutative and associative. So we can reorder the product such that we multiply the coefficients and variables separately. This gives us negative five times negative seven multiplied by 𝑚 raised to the ninth power times 𝑚 raised to the seventh power. We can recall that multiplying two negatives gives us a positive. So negative five times negative seven is equal to 35.
In the other factor, we have the product of exponential expressions. We can simplify this product by recalling the product rule for exponents. This tells us that 𝑥 raised to the power of 𝑎 times 𝑥 raised to the power of 𝑏 is equal to 𝑥 raised to the power of 𝑎 plus 𝑏. In other words, if we multiply exponential expressions with the same base, we just need to raise the base to the sum of the exponents.
We can apply this result with 𝑥 equal to 𝑚, 𝑎 equal to nine, and 𝑏 equal to seven to simplify the factor to be 𝑚 raised to the power of nine plus seven. We can then evaluate the expression in the exponent to obtain 35𝑚 raised to the 16th power. Hence, by applying the laws of exponents, we were able to simplify negative five times 𝑚 raised to the ninth power multiplied by negative seven 𝑚 raised to the seventh power to be 35𝑚 raised to the 16th power.