Simplify one-third 𝑥 raised to the
seventh power multiplied by four-fifths 𝑥 cubed.
In this question, we are asked to
simplify the product of two algebraic expressions. If we analyze the two factors, we
can see that they are the product of constants and variables that are raised to
nonnegative integer exponents. So both factors are monomials.
To simplify the product of
monomials, we first want to use the commutativity and associativity of
multiplication to rewrite the product so that we multiply the constants and
variables separately. This gives us one-third times
four-fifths multiplied by 𝑥 raised to the seventh power times 𝑥 cubed.
We can then evaluate each product
separately. First, we multiply fractions by
multiplying their numerators and denominators separately. We obtain one times four over three
times five. Second, we can simplify the product
of the variables by using the product rule for exponents. This tells us that if 𝑚 and 𝑛 are
nonnegative integers, then 𝑥 raised to the power of 𝑚 times 𝑥 raised to the power
of 𝑛 is equal to 𝑥 raised to the power of 𝑚 plus 𝑛. In other words, when multiplying
exponential expressions with the same base raised to nonnegative integer exponents,
we can instead raise the base to the sum of the exponents.
We can apply this result to
simplify our product with 𝑚 equal to seven and 𝑛 equal to three. We get 𝑥 raised to the power of
seven plus three. Finally, we can evaluate to obtain
four over 15 multiplied by 𝑥 raised to the 10th power.