Understanding the Cube Root of a Negative Number

The question asks us to find the cube root of -13824. A cube root is a number that, when multiplied by itself three times, gives the original number. The symbol for cube root is \(\sqrt[3]{}\).

An important property of cube roots is that the cube root of a negative number is always negative. For example, the cube root of -8 is -2 because \((-2) \times (-2) \times (-2) = 4 \times (-2) = -8\).

Therefore, since we are finding the cube root of -13824, which is a negative number, the answer must be a negative number. This eliminates options 1 (38) and 3 (24).

Finding the Cube Root of 13824 (Positive Part)

To find the cube root of -13824, we first find the cube root of its positive counterpart, 13824. We can use the prime factorization method for this.

Let's break down 13824 into its prime factors:

So, the prime factorization of 13824 is:

\(13824 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\)

We can write this in exponential form:

\(13824 = 2^9 \times 3^3\)

Calculating the Cube Root using Prime Factors

To find the cube root of 13824, we group the prime factors in sets of three and take one factor from each group. Alternatively, we can divide the exponents by 3.

\(\sqrt[3]{13824} = \sqrt[3]{2^9 \times 3^3}\)

Using the property \(\sqrt[3]{a^m} = a^{m/3}\), we get:

\(\sqrt[3]{13824} = 2^{9/3} \times 3^{3/3}\)

\(\sqrt[3]{13824} = 2^3 \times 3^1\)

Now, we calculate the values:

\(2^3 = 2 \times 2 \times 2 = 8\)

\(3^1 = 3\)

So,

\(\sqrt[3]{13824} = 8 \times 3 = 24\)

Final Cube Root Calculation

We found that the cube root of 13824 is 24. Since we are looking for the cube root of -13824, and the cube root of a negative number is negative, we simply take the negative of the positive result.

\(\sqrt[3]{-13824} = - \sqrt[3]{13824}\)

\(\sqrt[3]{-13824} = -24\)

Checking the Answer

To verify our answer, we can cube -24:

\((-24)^3 = (-24) \times (-24) \times (-24)\)

\((-24) \times (-24) = 576\)

\(576 \times (-24)\)

Let's perform the multiplication:

\(576 \times 24\)

• \(576 \times 4 = 2304\)
• \(576 \times 20 = 11520\)
• \(2304 + 11520 = 13824\)

So, \((-24) \times (-24) \times (-24) = 576 \times (-24) = -13824\).

This confirms that the cube root of -13824 is indeed -24.

Comparing this with the given options, we find that -24 is one of the options.

Revision Table: Key Concepts

Additional Information: Perfect Cubes and Cube Roots

A perfect cube is an integer that is the cube of another integer. Examples include 1 (\(1^3\)), 8 (\(2^3\)), 27 (\(3^3\)), 64 (\(4^3\)), 125 (\(5^3\)), 216 (\(6^3\)), etc.

Finding the cube root is the inverse operation of cubing a number. For instance, since \(4^3 = 64\), the cube root of 64 is 4. Similarly, since \((-5)^3 = -125\), the cube root of -125 is -5.

Knowing the cubes of the first few integers can be helpful in quickly estimating or finding cube roots:

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)
• \(6^3 = 216\)
• \(7^3 = 343\)
• \(8^3 = 512\)
• \(9^3 = 729\)
• \(10^3 = 1000\)
• \(20^3 = 8000\)
• \(30^3 = 27000\)

Our number, 13824, is between 8000 (\(20^3\)) and 27000 (\(30^3\)). The last digit of 13824 is 4. The last digit of a perfect cube's cube root determines the last digit of the cube itself. The only digit whose cube ends in 4 is 4 (\(4^3=64\)). This suggests the cube root of 13824 ends in 4. Numbers between 20 and 30 ending in 4 is 24. This simple trick could quickly point to 24 as the positive cube root.