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# Estimating Roots of any Number

### Estimating Roots of any Number

Some roots do not form whole numbers. However, we can exact roots that we know to estimate the value of any root.

What happens when numbers don't have a perfect root?

$30$ is not a square number, so finding the value of $\sqrt{30}$ is rather different. In fact, the value of $\sqrt{30}$ is most likely irrational (not able to be expressed as an exact fraction) and can't be expressed as an exact fraction.

Therefore, we can only **estimate** the value of $\sqrt{30}$. In other words, we can make a good guess at what the value would be.

Is $\sqrt{52}$ a rational number?

Which two integers will the value of $\sqrt{30}$ lie between?

Find the square numbers on either side of $30$

$25$ is the first square number less than $30$, and $36$ is the first square number more than $30$.

What is the integer value of $\sqrt{25}$?

What is the integer value of $\sqrt{36}$?

$\sqrt{30}$ lies between $\sqrt{25}$ and $\sqrt{36}$.

So this means it must lie between the integers $5$ and $6$.

Which two integers would $\sqrt[3]{200}$ lie between?

Find the cube numbers on either side of $200$.

$125$ is the first cube number less than $200$ and $216$ is the first cube number more than $200$.

What is the value of $\sqrt[3]{125}$?

What is the value of $\sqrt[3]{216}$?

$\sqrt[3]{200}$ must lie between $\sqrt[3]{125}$ and $\sqrt[3]{216}$

So that means $\sqrt[3]{200}$ lies between $5$ and $6$.

Which two integers would $\sqrt{68}$ lie between?

Sometimes, it's useful to estimate roots to a given degree of accuracy.

**Estimate** the value of $\sqrt6$ to 1 decimal place.

Find the integers this root would lie between

$\sqrt6$ should lie between $\sqrt4$ and $\sqrt9$. Therefore, it's between$2$ and $3$ .

$\sqrt6$ is roughly in the middle of $\sqrt4$ and $\sqrt9$

So we can estimate the value to be in the middle of $2$ and $3$,

This means we can say that $\sqrt6 \approx 2.5$

Well done!

**Estimate** the value of $\sqrt{80}$ to 1 decimal place.

$\sqrt{80}$ lies in between $8$ and which other integer?

$\sqrt{80}$ is between $\sqrt{64}$ and $\sqrt{81}$, but this time it is much closer to one than the other.

$\sqrt{80}$ is very close to $\sqrt{81}$

This means our estimate should be closer to $9$

So we may estimate $\sqrt{80} \approx 8.9$.

Which would be an appropriate estimate of $\sqrt{14}$?