Question: Cube Root of a Decimal

Comment on Cube Root of a Decimal

Way simpler than thought it to be.

((0.000008)^1/3 )^-1 = 1/((0.000008)^1/3 ) (1/((0.02*0.02*0.02)^(1/3)))
= 1/(( (0.02)^3 )^1/3) = 1/ ((0.02)^3/3) = 1/0.02
= 1/(2/100) = 100/2 = 50

How do we know the cube root of 1 million without a calculator?
greenlight-admin's picture

Good question.

We know that if ∛x = y, then x = y³ = (y)(y)(y)
For example, if ∛8 = 2, then 8 = 2³ = (2)(2)(2)

Likewise, if ∛(1,000,000) = y, then 1,000,000 = y³
= (y)(y)(y)

Now notice that, whenever we multiply powers of 10, we add the number of zeros.

For example, (100)(100,000) = 10,000,000
In other words, (power of 10 with TWO zeros)(power of 10 with FIVE zeros) = power of 10 with SEVEN zeros

Now notice that 1,000,000 has SIX zeros.
So, if we place TWO zeros in each set of brackets (above), we get:
(100)(100)(100), which equals 1,000,000

So, ∛(1,000,000) = 100

Does that help?

Cheers,
Brent

How do we know to re-write it as over the cubed root of 1,000,000? Why 1,000,000
greenlight-admin's picture

We can work our way to that as follows:
0.8 = 8/10
0.08 = 8/100
0.008 = 8/1,000
0.0008 = 8/10,000
0.00008 = 8/100,000
0.000008 = 8/1,000,000

Cheers,
Brent

Is there an alternative way to solve this without the fraction?
greenlight-admin's picture

You bet here's another solution:
We'll use the fact that ∛(ab) = (∛a)(∛b)

We'll calculate: [∛(0.000008)]^(-1)

Let's first deal with: ∛(0.000008)
Rewrite as: ∛(8 x 0.000001)
Rewrite as: ∛(8 x 10^-6)
Rewrite as: (∛8)(∛10^-6)
Evaluate: (2)(10^-2)

Now we're deal with the ^(-1) part
We now have: [(2)(10^-2)]^(-1)
Apply Power of a Product law: [2^(-1)][(10^-2)^(-1)]
Simplify: [2^(-1)][10^2]
Simplify: [1/2][100]
Simplify: 50

Cheers,
Brent

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