Find the cube root of 78402752

Question

Prepp Team · Accepted Answer

Finding the Cube Root of 78402752

The question asks us to find the cube root of the number 78402752. Finding the cube root means finding a number that, when multiplied by itself three times, equals 78402752. We can write this mathematically as finding \(x\) such that \(x^3 = 78402752\), or \(x = \sqrt[3]{78402752}\).

We can use a method involving the unit digit and estimation to solve this problem efficiently, especially when dealing with perfect cubes and multiple-choice options.

Step 1: Analyze the Unit Digit

The unit digit of the number 78402752 is 2. We need to find which digit, when cubed, results in a number with a unit digit of 2. Let's look at the cubes of single digits:

\(0^3 = 0\)
\(1^3 = 1\)
\(2^3 = 8\)
\(3^3 = 27\) (unit digit 7)
\(4^3 = 64\) (unit digit 4)
\(5^3 = 125\) (unit digit 5)
\(6^3 = 216\) (unit digit 6)
\(7^3 = 343\) (unit digit 3)
\(8^3 = 512\) (unit digit 2)
\(9^3 = 729\) (unit digit 9)

From the list above, we can see that only the digit 8, when cubed (\(8^3 = 512\)), results in a number with a unit digit of 2. Therefore, the unit digit of the cube root of 78402752 must be 8.

Step 2: Examine the Options Based on the Unit Digit

Let's look at the given options and check their unit digits:

Option 1: 534 (Unit digit is 4)
Option 2: 436 (Unit digit is 6)
Option 3: 428 (Unit digit is 8)
Option 4: 432 (Unit digit is 2)

Based on our analysis in Step 1, the unit digit of the correct cube root must be 8. Out of the given options, only 428 has a unit digit of 8.

Step 3: Verify the Potential Answer

Since only one option, 428, matches the required unit digit, it is highly likely to be the correct answer. To confirm, let's calculate the cube of 428:

\(428^3 = 428 \times 428 \times 428\)

First, calculate \(428 \times 428\):

		4	2	8
×		4	2	8
	3	4	2	4		(428 × 8)
		8	5	6		(428 × 20)
	1	7	1	2			(428 × 400)

	1	8	3	1	8	4	(Sum)

\(428 \times 428 = 183184\)

Now, calculate \(183184 \times 428\):

	1	8	3	1	8	4
×				4	2	8
	1	4	6	5	4	7	2	(183184 × 8)
		3	6	6	3	6	8		(183184 × 20)
	7	3	2	7	3	6			(183184 × 400)

	7	8	4	0	2	7	5	2	(Sum)

\(428 \times 428 \times 428 = 78402752\)

The result matches the original number. Therefore, the cube root of 78402752 is indeed 428.

Conclusion

Using the unit digit method was a quick way to narrow down the options. Verifying the identified option confirms that 428 is the correct cube root of 78402752.

\(\sqrt[3]{78402752} = 428\)