Solutions

1. \({\text{x}}^{4}\) (Written Solution)

   The product rule states that if two factors raised to an exponent are being multiplied together, and they have the same base, we can add the exponents.

   In this example, \({\text{x}}\) and \({\text{x}}^{3}\) are our two factors. Factors are the numbers that multiply together to make another number or expression.

   

   A number without an exponent is the same as a number with exponent 1.

   \({\text{x}} = {\text{x}}^{1}\)

   \({\color{Blue} {\text{x}}}^{{\color{Red} ?}}{\color{DarkGreen} {\text{x}}^{3}}\rightarrow {\color{Blue} {\text{x}}^{1}}{\color{DarkGreen} {\text{x}}^{3}}\)

   We can rewrite our example problem as \({\text{x}}^{1}{\text{x}}^{3}\).

   So \({\text{xx}}^{3} = {\text{x}}^{1}{\text{x}}^{3}\)

   In this example, both factors have the same base x, so we can add the exponents together.

   \( {\text{x}}^{1}{\text{x}}^{3} = {\text{x}}^{\left ( 1 + 3 \right )} = {\text{x}}^{4}\)

   Another way to look at this is to examine what the factors mean.

   \({\text{x}}^{3}\) Is the same as x multiplied together 3 times.

   \({\color{Blue} {\text{x}}}{\color{DarkGreen} {\text{x}}^{3}}= {\color{Blue} {\text{x}}}{\color{DarkGreen} {\text{xxx}}}\)

   So \({\text{xx}}^{3} = {\text{xxxx}}\)

   This is a total of 4 x’s multiplied together. We can rewrite that as \({\text{x}}^{4}\).

2. \({\text{m}}^{7}\)
3. \({\text{m}}^{4}{\text{x}}^{6}\) (Solution Video | Transcript)
4. \({\text{x}}^{6}{\text{y}}^{3}\)
5. \({\text{m}}^{4}{\text{x}}^{9}\) (Written Solution)

   Below there are two different ways to solve the same problem.

   Version 1:

   In this example, there are four factors: \({\text{m}}\), \({\text{x}}^{2}\), \({\text{m}}^{3}\), and \({\text{x}}^{7}\).

   

   We don’t see an exponent on the factor \({\text{m}}\), but this actually means it has an exponent of 1.

   

   Since everything is being multiplied together, we can rearrange the factors so the m factors are next to each other and the \({\text{x}}\) factors are next to each other.

   \({\text{m}}^{1}{\text{x}}^{2}{\text{m}}^{3}{\text{x}}^{7} = {\text{m}}^{1}{\text{m}}^{3}{\text{x}}^{2}{\text{x}}^{7}\)

   

   Using the product rule, we can add the exponents of the factors with the same base, so we can add the exponents of the \({\text{m}}\) factors and the exponents of the \({\text{x}}\) factors.

   \({\text{m}}^{1}{\text{m}}^{3}{\text{x}}^{2}{\text{x}}^{7} = {\text{m}}^{\left ( 1+3 \right )}{\text{x}}^{\left ( 2+7 \right )}\)

   

   Our final answer is \({\text{m}}^{4}{\text{x}}^{9}\)

   Version 2:

   Another way to look at this is to break all the factors with exponents into the multiplication of their bases.

   

   \({\text{m}}\) multiplied 1 time

   \({\text{x}}\) multiplied 2 times

   \({\text{m}}\) multiplied 3 times

   \({\text{x}}\) multiplied 7 times

   \({\text{mxxmmmxxxxxxx}}\)

   Since all the factors are being multiplied together we can rearrange them so the factors with the same base are next to each other.

   \({\text{mxxmmmxxxxxxx = mmmmxxxxxxxxx}}\)

   

   We see there are 4 m factors being multiplied together and 9 \({\text{x}}\) factors being multiplied together. The final step is to rewrite this in exponent form.

   Our final answer is \({\text{m}}^{4}{\text{x}}^{9}\)

6. \({\text{x}}^{7}{\text{y}}^{5}{\text{z}}^{2}\) (Solution Video | Transcript)