SWBAT simplify expressions using the Power of a Power property

When powers of powers are calculated, interesting patterns emerge.

15 minutes

Hand each student an APK ENTRANCE SLIP as they enter the classroom.

*APK ENTRANCE SLIP*

*1. If (goof) ^{3} = (goof) (goof) (goof), what is (5^{x})^{3}?*

*"student should write (5^{x})(5^{x})(5^{x})"*

*2. Using the product of powers property, write your answer to question 1 as a single power.*

*"students should write 5 ^{x+x+x }or 5^{3x "}*

*3. Analyze the answers to question 1 and 2 and make a verbal conjecture about the power of a power property.*

We want to access student's prior knowledge (APK) and use repeated multiplication for powering. Then use the Product of Powers Property to explain the Power of a Power Property. I always find that going back to the meaning of powers as repeated multiplication is never a waste of time.

Walk around observing student's work. Some students may go too far out of what we want them to figure out. Once students are done, ask someone to share their work on the board for all to see.

30 minutes

The general pattern shown in the Launch exercise is the Power of a Power Property.

Ask the students to use the same pattern (using repeated multiplication and the product of powers property) to write **(5 ^{2})^{4}** as a single power.

Ask them to do it at their desks, then compare and discuss their work with their elbow partner, before calling a student to the board to show what they wrote.

Students are expected to write; (5^{2})^{4} **= **5^{2} · 5^{2} · 5^{2} · 5^{2} **= **5 ^{2 }^{+ }^{2 }^{+ }^{2 }^{+ }^{2} **= **5^{8}

Once this is done, write the general rule on the board:

**(b ^{m})^{n} = b^{mn} for all values m and n; b‡0**

Then tell the class that sometimes expressions involve both powers of powers and multiplication and write the following example on the whiteboard;

**Simplify 3 m(m^{4})^{2}**

I would allow them to try this on their own.

A common mistake is that they may take both m and *m ^{4}* to the second power and write 3m

*3m ∙ m ^{4} ∙ m^{4}* or

In both cases resulting in *3m ^{9}*

I would ask the class here to signal with fingers from 1 to 5; 5 for best understanding, just to get an idea of how well they think they've understood this last section. If someone signals low understanding, I would work with these students together in a group while the rest do the following problems.

Write the following problems on the board for students to perform:

**Rewrite the expression as a single power. **

**1. 2(k ^{10})^{7} 2. 4k^{2}(k^{3})^{5} **

**Find the value of n in each case:**

**3. (5 ^{6})^{x} = 5^{6 }4. (a^{7} · a^{n})^{2} = a^{24}**

**Call on four students to the board to show their work. **

5 minutes

**Notebook Entry: **Ask students to write 2 key ideas they learned today in their notebooks. Tell them that students will be called at random to share what they wrote to launch our next lesson.