Understanding the Square Root of 4096

The question asks us to find the square root of the number 4096. The square root of a number is a value that, when multiplied by itself, gives the original number. If 'x' is the square root of 'y', it means that \(x^2 = y\), or \(x \times x = y\). We are looking for a number 'x' such that \(x \times x = 4096\).

Methods to Find the Square Root of 4096

There are several ways to find the square root of a number like 4096. Two common methods are prime factorization and estimation followed by verification.

Using Prime Factorization

Prime factorization involves breaking down the number into its prime factors. For 4096, we can repeatedly divide by the smallest prime number, which is 2.

• \(4096 \div 2 = 2048\)
• \(2048 \div 2 = 1024\)
• \(1024 \div 2 = 512\)
• \(512 \div 2 = 256\)
• \(256 \div 2 = 128\)
• \(128 \div 2 = 64\)
• \(64 \div 2 = 32\)
• \(32 \div 2 = 16\)
• \(16 \div 2 = 8\)
• \(8 \div 2 = 4\)
• \(4 \div 2 = 2\)
• \(2 \div 2 = 1\)

So, the prime factorization of 4096 is \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\). This is \(2^{12}\).

To find the square root, we group the prime factors in pairs:

\(\sqrt{4096} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)}\)

For each pair of identical factors under the square root sign, we take out one factor:

\(\sqrt{4096} = 2 \times 2 \times 2 \times 2 \times 2 \times 2\)

Multiplying these together:

\(2 \times 2 = 4\)
\(4 \times 2 = 8\)
\(8 \times 2 = 16\)
\(16 \times 2 = 32\)
\(32 \times 2 = 64\)

Thus, the square root of 4096 is 64.

Verifying the Answer by Squaring

To check if 64 is indeed the square root of 4096, we can multiply 64 by itself:

\(64 \times 64\)

Or, using standard multiplication:

 64
\(\underline{\times 64}\)
 256 (64 \(\times\) 4)
3840 (64 \(\times\) 60)
4096

Since \(64 \times 64 = 4096\), the square root of 4096 is confirmed to be 64.

Examining the Options

Let's briefly look at the other options:

• Option 1: 68. \(68 \times 68 = 4624\). This is not 4096.
• Option 3: 74. \(74 \times 74 = 5476\). This is not 4096.
• Option 4: 56. \(56 \times 56 = 3136\). This is not 4096.

Only 64, when squared, gives 4096.

Revision Table: Square Roots of Perfect Squares

Additional Information on Square Roots and Perfect Squares

A number like 4096 is called a perfect square because its square root is a whole number (64). Not all numbers are perfect squares (e.g., 2, 3, 5, etc., have square roots that are not whole numbers).

The symbol for the square root is \(\sqrt{}\), called a radical sign. So, \(\sqrt{4096} = 64\).

Every positive number has two square roots, one positive and one negative. For example, both \(64 \times 64 = 4096\) and \((-64) \times (-64) = 4096\). However, when the question asks for "the square root" without specifying positive or negative, it usually refers to the principal (positive) square root.