Calculating the Sixth Root of 3

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In summary, cbrt{3} x cbrt{3} = 3^(1/3) * 3^(1/3) = 3^(2/3) = 3^(1/3 + 1/3) = 3^(1/3) * 3^(1/3) = sqrt[3]{3} * sqrt[3]{3}, and the correct answer is 2/3.
  • #1
mathdad
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Let cbrt = cube rootcbrt{3} x cbrt{3} =

(3)^(1/3) * (3)^(1/3)

3^(1/6) ir sixth root {3}

Correct?
 
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  • #2
In general, you want to use the following rule:

\(\displaystyle \sqrt[3]{a}\cdot\sqrt[3]{b}=\sqrt[3]{ab}\)

Using exponents, we can get the same result:

\(\displaystyle a^c\cdot b^c=(ab)^c\)

Now, when the base is the same, we can simply add exponents:

\(\displaystyle a^b\cdot a^c=a^{b+c}\)

So, in the given expression, we may write:

\(\displaystyle \sqrt[3]{3}\cdot\sqrt[3]{3}=\sqrt[3]{3\cdot3}=\sqrt[3]{3^2}=3^{\frac{2}{3}}=3^{\frac{1}{3}+\frac{1}{3}}=3^{\frac{1}{3}}\cdot3^{\frac{1}{3}}=\sqrt[3]{3}\cdot\sqrt[3]{3}\)
 
  • #3
MarkFL said:
In general, you want to use the following rule:

\(\displaystyle \sqrt[3]{a}\cdot\sqrt[3]{b}=\sqrt[3]{ab}\)

Using exponents, we can get the same result:

\(\displaystyle a^c\cdot b^c=(ab)^c\)

Now, when the base is the same, we can simply add exponents:

\(\displaystyle a^b\cdot a^c=a^{b+c}\)

So, in the given expression, we may write:

\(\displaystyle \sqrt[3]{3}\cdot\sqrt[3]{3}=\sqrt[3]{3\cdot3}=\sqrt[3]{3^2}=3^{\frac{2}{3}}=3^{\frac{1}{3}+\frac{1}{3}}=3^{\frac{1}{3}}\cdot3^{\frac{1}{3}}=\sqrt[3]{3}\cdot\sqrt[3]{3}\)

Great but is my answer wrong?
 
  • #4
RTCNTC said:
Great but is my answer wrong?

Yes, your result is incorrect. :D

You want to add the two exponents to get 1/3 + 1/3 = 2/3.
 
  • #5
MarkFL said:
Yes, your result is incorrect. :D

You want to add the two exponents to get 1/3 + 1/3 = 2/3.

I forgot that powers are added.
 
  • #6
Well, what in the world did you do to get "1/6"?
 
  • #7
Thank you everyone.
 

FAQ: Calculating the Sixth Root of 3

What is Cube Root x Cube Root?

Cube Root x Cube Root is a mathematical expression that represents the product of two cube roots. It can be written as ∛x * ∛x, where ∛x represents the cube root of x.

How do you simplify Cube Root x Cube Root?

To simplify Cube Root x Cube Root, you can use the property of exponents which states that (x^m) * (x^n) = x^(m+n). In this case, the cube roots can be multiplied together to get the cube root of x^2, which is equal to ∛(x^2). This can also be written as x^(2/3).

What is the relationship between Cube Root x Cube Root and x squared?

The relationship between Cube Root x Cube Root and x squared is that they are equivalent expressions. Both represent the cube of x, or x raised to the power of 3.

Can Cube Root x Cube Root be negative?

Yes, Cube Root x Cube Root can be negative. The cube root of a negative number is also negative, so when two negative cube roots are multiplied together, the result is a negative number.

How is Cube Root x Cube Root used in real life?

Cube Root x Cube Root can be used in various real-life situations, such as calculating the volume of a cube or finding the length of one side of a cube given its volume. It can also be used in engineering and physics to solve problems involving cubic equations and functions.

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