Solutions

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

   The power rule for exponents says that any base raised to an exponent that is also raised to an exponent is the same as that base raised to the multiplication of the two exponents.

   In this example, x is the base. It is being raised first to the power of m and then all of that is being raised to the power of n.

   According to the power rule for exponents, this is the same as:

   \(({\text{x}}^{\text{m}})^{\text{n}} = {\text{x}}^{\text{mn}}\)

   where m and n are being multiplied together.

2. \(3^{6} = 729\)
3. \({\text{m}}^{8}{\text{y}}^{10}\) (Written Solution)

   Version 1: Applying the power rule for exponents

   \(\left ({\text{m}}^{2} \right )^{4}\left ({\text{y}}^{5} \right )^{2}\)

   According to the power rule for exponents, we can multiply the \(2 \cdot 4\) to get the exponent for m. We can also multiply the \(5 \cdot 2\) to get the exponent for y.

   \(\left ( {\text{m}}^{2} \right )^{4}\left ( {\text{y}}^{5} \right )^{2} = {\text{m}}^{\left ( 2\cdot4 \right )}{\text{y}}^{\left ( 5\cdot2 \right )} = {\text{m}}^{8}{\text{y}}^{10}\)

   Our final answer is \({\text{m}}^{8}{\text{y}}^{10}\).

   Version 2: Applying the rules of multiplication to show why the power rule for exponents works

   \(\left ({\text{m}}^{2} \right )^{4}\left ({\text{y}}^{5} \right )^{2}\)

   This means \({\text{m}}^{2}\) is being multiplied together 4 times and \({\text{y}}^{5}\) is being multiplied together 2 times.

   We can multiply it out like this:

   \({\text{m}}^{2}{\text{m}}^{2}{\text{m}}^{2}{\text{m}}^{2}{\text{y}}^{5}{\text{y}}^{5}\)

   \(\left ( {\color{DarkGreen} {\text{m}}^{2}} \right )^{4}\left ( {\color{Blue} {\text{y}}^{5}} \right )^{2} = {\color{DarkGreen} {\text{m}}^{2}{\text{m}}^{2}{\text{m}}^{2}{\text{m}}^{2}}{\color{Blue} {\text{y}}^{5}{\text{y}}^{5}}\)

   Each factor \({\text{m}}^{2}\) is the same as \({\text{m}}\) multiplied twice. Each \({\text{y}}^{5}\) is the same as \({\text{y}}\) multiplied 5 times.

   If we expand out each factor using the rules of multiplication it becomes the following:

   \({\text{m m m m m m m m y y y y y y y y y y}}\)

   

   We now see there are 8 m’s being multiplied together and 10 y’s being multiplied together. If we write this in exponent form it is \({\text{m}}^{8}{\text{y}}^{10}\).

   

   Our final answer is \({\text{m}}^{8}{\text{y}}^{10}\), which is the same answer as when we used the power rule.

4. \({\text{x}}^{8}{\text{y}}^{8}\) (Solution Video | Transcript)
5. \(4^{6}2^{5} = 4096 \cdot 32 = 131,072\) (Solution Video | Transcript)
6. \({\text{a}}^{2{\text{x}}}{\text{b}}^{6} \)