What is the cube root of 1/4?
Understand the Problem
The question is asking for the cube root of the fraction 1/4. This involves finding a number that, when multiplied by itself three times, results in 1/4.
Answer
The cube root of $\frac{1}{4}$ is $\frac{1}{\sqrt[3]{4}}$.
Answer for screen readers
The cube root of $\frac{1}{4}$ is $\frac{1}{\sqrt[3]{4}}$.
Steps to Solve
- Set the Equation for Cube Root
To find the cube root of $\frac{1}{4}$, we can express it mathematically as:
$$ x = \sqrt[3]{\frac{1}{4}} $$
This means we are looking for a value of $x$ such that $x^3 = \frac{1}{4}$.
- Rewrite the Fraction
Next, we can rewrite $\frac{1}{4}$ as:
$$ \frac{1}{4} = \frac{1^3}{2^2} $$
Using this, we can re-express the cube root:
$$ x = \sqrt[3]{\frac{1^3}{2^2}} $$
- Apply the Cube Root to the Numerator and Denominator
Now, we calculate the cube root separately for the numerator and the denominator:
$$ x = \frac{\sqrt[3]{1^3}}{\sqrt[3]{2^2}} $$
Since $\sqrt[3]{1^3} = 1$ and $\sqrt[3]{2^2} = 2^{2/3}$, we have:
$$ x = \frac{1}{2^{2/3}} $$
- Rewrite the Denominator
The expression $2^{2/3}$ can also be represented with a radical:
$$ 2^{2/3} = \sqrt[3]{2^2} = \sqrt[3]{4} $$
So, we can write:
$$ x = \frac{1}{\sqrt[3]{4}} $$
- Final Expression for the Cube Root
The final answer for the cube root of $\frac{1}{4}$ is:
$$ x = \frac{1}{\sqrt[3]{4}} $$
The cube root of $\frac{1}{4}$ is $\frac{1}{\sqrt[3]{4}}$.
More Information
The cube root of a fraction gives insight into dividing values evenly. In real-life applications, such as physics and engineering, cube roots can help find dimensions of objects with a specific volume.
Tips
- Confusing cube roots with square roots. Make sure to identify the correct root needed for the problem.
- Not simplifying the fraction correctly before finding the root. Always express fractions in their simplest forms.
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