Aproximação

Approximating Cube Roots

Here is a three-step way to approximate cube roots.

Step One

Identify the next highest and lowest perfect cubes around the number you are taking the cube root of. For example, say you are trying to find $\sqrt[3]{2}.$ The next highest and lowest perfect cubes around $2$ are $8$ and $1$. Notice that $2$ is closer to $1$ than it is to $8.$

Step Two

Take the cube roots of all three numbers and simplify. We will compare all of the cube roots with inequalities. $$1 < 2 < 8$$ $$\Rightarrow \sqrt[3]{1} < \sqrt[3]{2} < \sqrt[3]{8}$$ $$\Rightarrow 1 < \sqrt[3]{2} < 2$$

Step Three

Approximate! Since $2$ is closer to $1$ than it is to $8,$ $\sqrt[3]{2}$ is closer to $\sqrt[3]{1}=1$ than it is to $\sqrt[3]{8}=2.$ Thus, we can approximate: $$\sqrt[3]{2} \approx 1.1$$

This is a pretty good approximation! If we use a [cube root] calculator, we can see that $\sqrt[3]{2} = 1.25992104989\dots$ The tenths place of this approximation and of $1.1$ differ only by $1$.

Here's a helpful visual for what we've gone over. We use approximating $\sqrt[3]{17}$ as another example.

Indeed, if you use a calculator, you can see that $\sqrt[3]{17} = 2.57128....$ Our approximating technique worked! Let's do another example.

Step One

Identify the next highest and lowest perfect cubes around $110.$ After some thought, we can determine that these are $64$ and $125.$ Notice that $110$ is closer to $125$ than it is to $64.$ To figure this out, we can subtract: $125 - 110 = 15$ but $110 - 64 = 46$.

Step Two

Set up an inequality, then take the cube roots of all three numbers and simplify. $$64 < 110 < 125$$ $$\Rightarrow \sqrt[3]{64} < \sqrt[3]{110} < \sqrt[3]{125}$$ $$\Rightarrow 4 < \sqrt[3]{110} < 5$$

Step Three

Approximate! Since $110$ is closer to $125$ than it is to $64, \sqrt[3]{110}$ is closer to $\sqrt[3]{125}=5$ than it is to $\sqrt[3]{64}=4.$ Thus, we decide to guess: $$\sqrt[3]{110} \approx 4.8$$

This is a pretty good approximation! If we use a calculator, we can see that $\sqrt[3]{110} = 4.79141985706\dots$

The $3$rd power of the cube root of a number makes that number

When a number is cubed, you take that number to the third power (in other words, multiply that number by itself $3$ times). For example, $8$ cubed is $8^3 = 8\times8\times8 = 512$. Because of this, you could also say that $8$ is the cube root of $512$, or that $\sqrt[3]{512} = 8$.

How to approximate the cube root of a number

The best way to approximate a cube root is a lot like the best way to approximate the square root: Just narrow it down! Let's say we've got the number $164$, and we want to find the cube root ($\sqrt[3]{164}$). How can we begin? Think about it for a second, and then keep reading.

Well, we know that $1^3 = 1$, since anything multiplied by $1$ stays the same. We also know that $8^3 = 512$, since we just did that above. So, $\sqrt[3]{164}$ has to be somewhere between $1$ and $8$. Take a second to think about what it might be, then keep reading on.

$164$ is a lot less than $512$, so $\sqrt[3]{164}$ is probably a lot smaller than $8$. Let's try guessing $4$.

$$4^3 = 4\times4\times4 = 64$$

Hm, that's a bit low. Let's try $6$.

$$6^3 = 6\times6\times6 = 216$$

$216$ is more than $164$, so we know that $\sqrt[3]{164}$ is more than $4$ and less than $6$. Let's try $5$.

$$5^3 = 5\times5\times5 = 125$$

Notice $125$ is still less than $164$, so the cube root of $164$ must be somewhere between $5$ and $6$. You can keep using this method to get closer, but that's all we need to approximate the value. Nice job!