Question Video: Simplifying the Product of Monomials Using the Laws of Exponents Mathematics

Video Transcript

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.