Find the cube roots of the following by finding the digits in the units and tens places

Options

A. 110592
B. 32768
C. 2744
D. n/a

Answer

To find the cube roots of the numbers given, we will follow these steps:

### Step 1: Identify the number
We will start with the first number, which is **110592**.

### Step 2: Determine the unit digit
The unit digit of **110592** is **2**. We need to find which number cubed gives a unit digit of **2**.

### Step 3: Find the cube root of the unit digit
The possible unit digits for cubes are:
- \(0^3 = 0\)
- \(1^3 = 1\)
- \(2^3 = 8\)
- \(3^3 = 27\)
- \(4^3 = 64\)
- \(5^3 = 125\)
- \(6^3 = 216\)
- \(7^3 = 343\)
- \(8^3 = 512\)
- \(9^3 = 729\)

From this, we see that the cube root of **2** corresponds to the cube of **8** (since \(8^3 = 512\), and the unit digit is **2**).

### Step 4: Remove the unit digit and analyze the remaining number
Now, we will look at the remaining digits after removing the unit digit. The remaining part of **110592** is **110**.

### Step 5: Find the cube root of the remaining number
We need to find the cube root of **110**. We will check the cubes of numbers:
- \(4^3 = 64\) (too small)
- \(5^3 = 125\) (too large)

Since \(4^3 < 110 < 5^3\), we can conclude that the cube root of **110** is between **4** and **5**.

### Step 6: Combine the results
Since the unit digit corresponds to **8** and the remaining part gives us a cube root of approximately **4**, we can conclude that the cube root of **110592** is approximately **48**.

### Step 7: Repeat for the next number
Now, let's find the cube root of the second number, **32768**.

### Step 8: Identify the unit digit
The unit digit of **32768** is **8**. The cube root of **8** is **2** (since \(2^3 = 8\)).

### Step 9: Remove the unit digit and analyze the remaining number
Removing the unit digit gives us **3276**.

### Step 10: Find the cube root of the remaining number
Now we check the cubes:
- \(14^3 = 2744\) (too small)
- \(15^3 = 3375\) (too large)

Since \(14^3 < 3276 < 15^3\), we can conclude that the cube root of **32768** is approximately **14**.

### Step 11: Move to the next number
Now, we will find the cube root of **2744**.

### Step 12: Identify the unit digit
The unit digit of **2744** is **4**. The cube root of **4** corresponds to **4** (since \(4^3 = 64\)).

### Step 13: Remove the unit digit and analyze the remaining number
The remaining part is **274**.

### Step 14: Find the cube root of the remaining number
Now we check the cubes:
- \(6^3 = 216\) (too small)
- \(7^3 = 343\) (too large)

Since \(6^3 < 274 < 7^3\), we can conclude that the cube root of **2744** is approximately **14**.

### Final Results
Thus, the cube roots of the numbers are:
- Cube root of **110592** is **48**.
- Cube root of **32768** is **14**.
- Cube root of **2744** is **14**.

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