Completing Chain of Inequalities with Cube Roots
The given mathematical statement to complete is a chain of inequalities featuring a cube root:
\[ \underline{\hspace{1cm}} < \sqrt[3]{107} < \underline{\hspace{1cm}} \]
To solve this, we need to find cube numbers that are immediately below and above 107, because the cube root of 107 will be between the cube roots of those numbers.
The nearest perfect cubes below 107 are:
\[ 4^3 = 64 \]
\[ 5^3 = 125 \]
Since \(125 > 107\), we've gone too far. The largest perfect cube less than 107 is \(4^3 = 64\). Therefore, the cube root of 64, which is 4, is the largest whole number cube root less than \( \sqrt[3]{107} \).
Now, looking above 107, \(5^3 = 125\) is the smallest perfect cube greater than 107. Therefore, the cube root of 125, which is 5, will be the smallest whole number cube root greater than \( \sqrt[3]{107} \).
So the complete statement is:
\[ 4 < \sqrt[3]{107} < 5 \]
