**Middle School Number Sense Lesson 62: Multiplying Powers of 2 and 5**

Today's lesson is an oldie but a goodie--multiplying some power of 2 by some power of 5. It allows us to turn the Order of Operations inside out. This concept appeared

**5 times**last year, with a median placement at

**question # 63**.

**Please Like our Facebook Page (**

**https://www.facebook.com/numberdojo/**

**) if you want to see new posts on your wall. I will reward you with a free concept index or flashcard file of your choice. Thank you!**

**Number Dojo Level: 257**

The secret to solving these problems is to factor out equal powers of 2 and 5. Here are the steps:

- Factor out the highest power shared by 2 and 5.
- Combine the
**2 x 5**to get**10**, and**then**raise 10 to that same power. Write down the appropriate number of 0's. - Calculate the leftover powers of 2 or 5, along with any other exponents, and write this in front.

**Example 1:**

**2**

^{5}x 5^{3}= ___1. The highest power shared by both 2 & 5 is

**3**. So we have

**2**.

^{2}x (2^{3}x 5^{3})2. Combine the

**2 x 5**to get

**2**. Write down

^{2}x (10^{3})**000**.

3. We're left with 2

^{2}. Write a

**4**in front. The answer is

**4000**.

**Example 2:**

**5**

^{3}x 2^{7}= ___1. The highest power shared by both 2 & 5 is

**3**. So we have

**(5**.

^{3}x 2^{3}) x 2^{4}2. Combine the

**5 x 2**to get

**(10**. Write down

^{3}) x 2^{4}**000**.

3. We're left with 2

^{4}. Write a

**16**in front. The answer is

**16000**.

**Example 3:**

**4**

^{3}x 25^{2}= ___1. Change 4

^{3}to

**2**and 25

^{6}^{2}to

**5**.

^{4}2. The highest power shared by both 2 & 5 is

**4**. So we have

**2**.

^{2}x (2^{4}x 5^{4})3. Combine the

**2 x 5**to get

**2**. Write down

^{2}x (10^{4})**0000**.

4. We're left with 2

^{2}. Write a

**4**in front. The answer is

**40000**.

**Example 4:**

**2**

^{3}x 5^{3}x 9^{3}= ___1. The highest power shared by both 2 & 5 is

**3**. So we have

**(2**.

^{3}x 5^{3}) x 9^{3}2. Combine the

**2 x 5**to get

**(10**. Write down

^{3}) x 9^{3}**000**.

3. We're left with 9

^{3}. Write a

**729**in front. The answer is

**729000**.

**Example 5:**

**How many digits are in 2**

^{8}x 5^{6}x 7^{2}? ___1. The highest power shared by both 2 & 5 is

**6**. So we have

**2**.

^{2}x (2^{6}x 5^{6}) x 7^{2}2. Combine the

**2 x 5**to get

**2**. We have

^{2}x (10^{6}) x 7^{2}**6**zeros.

3. We're left with 2

^{2}x 7

^{2}, which is 14

^{2}, which is

**196**. That's

**3**more digits, so the answer is

**9**.

**Here's a free worksheet to help you practice MultPowers2&5:**

multpowers2_5.pdf |

**Up Next for Middle School: TriangularNum**