**High School Number Sense Lesson 50: Estimating with Powers Higher than 2**

Estimation questions are often missed on number sense tests. I think it is difficult to learn to round accurately (is that an oxymoron?)... This concept combines our understanding of exponents (which we haven't covered yet in this blog) and multiplication. It appeared

**16 times**last year--8 times each at

**question # 50**and

**question # 60**.

**Number Dojo Level: 277**

On middle school tests, students are used to seeing a single integer or decimal, usually less than 10, raised to some power. We will cover this variation in the first few examples. But on high school tests, students are often asked to multiply and/or divide 2 or more such numbers. And like I always teach, when asked to multiply and divide in the same problem,

**divide before you multiply!**

**Example 1:**

*** 14**

^{3}= ___1. 14

^{3}is the same as 14 x 14 x 14. By now, hopefully you have memorized the fact that 14 x 14 =

**196**. So really all we need to do is multiply 196 x 14.

2. We can safely round 196 to 200, and multiply that by 14 to get

**2800**.

3. The exact answer is

**2744**, and the acceptable (+/- 5%) range is

**from 2607 to 2881**. We are safely within range.

**Example 2:**

*** 9**

^{4}= ___1. 9

^{4}= 9 x 9 x 9 x 9.

2. 9 x 9 =

**81**, so we need to multiply 81 x 81.

3. We can round 81 to 80 and then square that to get

**6400**.

4. The exact answer is

**6561**, and the acceptable (+/- 5%) range is

**from 6233 to 6889**. We are safely within range.

**Example 3:**

*** 6.7**

^{4}= ___1. First let's estimate 6.7

^{2}. 6.7 is basically 2/3 of 10, so we could say 6.7 x 2/3 x 10.

2. 2/3 x 6.7 is about

**4.4**. 4.4 x 10 =

**44**.

3. Now we need to multiply 44 x 44. (There is a shortcut to that, which I'll probably teach later). We can use

**Double/Half**to make this 22 x 88, and then again to make it

**11 x 176**.

4. Using

**Mult11**, we can calculate 11 x 176 to be

**1936**.

5. The exact answer is

**2015.1121**, and the acceptable (+/- 5%) range would be

**from 1915 to 2115**. We are on the low side, but safely within range.

**Example 4:**

*** 2.3**

^{6}= ___1. We have already memorized that 23

^{2}=

**529**, which means that 2.3

^{2}=

**5.29**. We can round this to

**5.3**.

2. We are left with

**5.3**, or

^{3}**5.3 x 5.3 x 5.3**.

3. FOILing, we can quickly determine that 5.3 x 5.3 =

**28.09**, which we can round down to 28.

4. Now we have to calculate

**5.3 x 28**. Using

**Double/Half**, we can change this to

**10.6 x 14**, which is the same as

**140 + 8.4**, which is about

**148**.

5. The exact answer is

**148.035889**, so we

__nailed it__.

**Example 5:**

*** 9**

^{3}x 27^{2}= ___1. 9

^{3}= 9 x 9 x 9 = 81 x 9 =

**729**. 27

^{2}is also

**729**.

2. So we have 729 x 729. We can round this to 730 x 730, or even better, round one up to 750 and the other down to 700.

3. 700 x 750 = 7 x 75 x 1000 =

**525000**.

4. The exact answer is

**531441**, and the acceptable range is

**from 504869 to 558013**.

**Example 6:**

*** 12**

^{4}÷ 4^{4}x 7^{2}= ___1. Notice that

**12**is divisible by

^{4}**4**. Divide 12

^{4}^{4}by 4

^{4}to get

**3**.

^{4}2. We are left with

**3**. 3

^{4}x 7^{2}^{4}=

**81**and 7

^{2}=

**49**. So we have

**81 x 49**.

3. Round 81 down to 80 and 49 up to 50. 80 x 50 =

**4000**.

4. The exact answer is

**3969**, and the acceptable range is

**from 3771 to 4167**.

**Here's a free worksheet to help you practice EstHighPower:**

esthighpower.pdf |

**Up Next for High School: Permutation**