Example Question - perfect cubes
Here are examples of questions we've helped users solve.
Approach to Finding Closest Integer to Cube Root
To solve the question in the image, we need to find the cube root of 572 and then identify the closest integer to this value. Calculating the exact cube root of 572 might be challenging without a calculator, but we can estimate the closest integer by finding the cube roots of perfect cubes near 572. The cube of 8 is \(8^3 = 512\), and the cube of 9 is \(9^3 = 729\). Since 572 falls between 512 and 729, the cube root of 572 will be between 8 and 9. Since 572 is closer to 512 than it is to 729, we can safely say that the cube root of 572 will be closer to 8 than to 9. Therefore, the integer closest to the cube root of 572 is 8.
Calculating Cube Root and Nearest Integer Estimation
To solve the cube root of 112 and find the integer closest to this value, we will first identify the perfect cubes near 112 to estimate the cube root. The perfect cubes closest to 112 are: - \( 4^3 = 64 \) - \( 5^3 = 125 \) Since 112 is between 64 and 125, we know the cube root of 112 will be between 4 and 5. Because 112 is closer to 125 than it is to 64, the cube root of 112 will be closer to 5. Therefore, the integer closest to \( \sqrt[3]{112} \) is 5.
Determining Closest Integer to Cube Root
The image is asking for the integer closest to the cube root of 111. To solve this, we should look for two perfect cubes that are close to 111 and then find their cube roots. We know that: - \(4^3 = 64\) - \(5^3 = 125\) Since 111 is between 64 and 125, the cube root of 111 must be between 4 and 5. As 111 is closer to 125 than to 64, the cube root of 111 is closer to 5 than to 4. Hence, the closest integer to \(\sqrt[3]{111}\) is 5.
Finding the Closest Cube Root to 111
The image contains a question asking which integer is closest to the cube root of 111. To answer this, let's look for perfect cubes near 111 and find their cube roots. - The perfect cube closest to 111 that is less than it is \( 4^3 = 64 \). - The next perfect cube greater than 111 is \( 5^3 = 125 \). Since 111 is closer to 125 than to 64, we can say that the cube root of 111 is closer to 5 than to 4. Thus, the integer closest to the cube root of 111 is 5.
Identifying Closest Integer to Cube Root
The question is asking us to find which integer is closest to the cube root of 55. To get an idea of which integers might be close, let's consider the perfect cubes near 55. The cube of 3 is \(3^3 = 27\), and the cube of 4 is \(4^3 = 64\). Since 55 lies between 27 and 64, the cube root of 55 must lie between 3 and 4. Since 55 is closer to 64 than it is to 27, the cube root of 55 will be closer to 4 than to 3. Therefore, the integer closest to the cube root of 55 is 4.
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: \[ 4 < \sqrt[3]{107} < 5 \]
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