Square Root of 343

In Mathematics, the square root of a number is a number which on multiplication by itself results in the original number. Thus, the square root of 343 is a number, which when multiplied by itself, should produce the result 343. In this article, we are going to discuss the value of the square root of 343, and also learn how to find the square root of 343 using the long division method and prime factorization method with complete explanation.

Table of Contents:

• What Is The Square Root Of 343?
• Square Root Of 343 In Radical Form
• Square Root Of 343 By Prime Factorization Method
• Square Root Of 343 By Long Division Method
• Examples
• FAQs

What is the square Root of 343?

The square root of 343 is a number, which when multiplied by itself gives the original number 343. The square root of 343 is symbolically represented as √343.

(i.e) = √343 = √(Number × Number)

Thus, if we multiply the number 18.520 for two times, we will get the number 343.

Therefore, √343 = √(18.520 × 18.520)

√343 = √(18.520)²

Now, remove square and square root on the right hand side, we get

√343 = ± 18.520.

Square Root of 343 in Radical Form

The square root of 343 can also be represented in the radical form. Since 343 is a perfect cube number, the number 343 is written as 7 × 7 × 7. Thus, the square root of 343 in radical form is written as follows:

√343 = √(7 × 7 × 7)

√343 = √(7) ×√(7)²

√343 = 7√7.

Therefore, the square root of 343 in radical form is 7√7.

Square Root of 343 by Prime Factorization Method

The square root of 343 can be found using two different methods, such as the prime factorization method and the long division method. Now, let us discuss the prime factorization method to find the square root of 343.

Let us consider the prime factorization of 343, 7³, which is equal to 7 × 7 × 7.

Thus, √343 = √(7 × 7 × 7) = 7√7, which is the radical form of the square root of 343.

To obtain the square root of 343 in decimal form, substitute the approximate value of the square root of 7 in the radical form.

Thus, √343 = 7√7

√343 = 18.520.

Therefore, the value of the square root of 343 in decimal form is 18.520 (approximately).

Square Root of 343 by Long Division Method

The value of the square root of 343 can be exactly found using the long division method. Generally, the long division method is used to find the value of the square root of a number, when a number is a non-perfect square number. Now, let us discuss how to find the value of the square root of 343 using the long division method.

Step 1: Write the number 343 in decimal form. To find the exact value of the square root of 343, add 6 zeros after the decimal point. Hence, 343 in decimal form is 343. 000000. Now, pair the number 343 from right to left by putting the bar on the top of the number.

Step 2: Now, divide the number 3 by a number, such that the product of the number (i.e. number×number) should be less than or equal to 3. Thus, 1×1 =1, which is less than 3. Thus, we obtained the quotient = 1 and remainder = 1.

Step 3: Double the quotient value, so we get 2, and assume that 20 is the new divisor. Now, bring down the value 43 for division operation. So, the new dividend obtained is 243. Now, find the number, such that (20 + new number) × new number should give the product value, that should be less than or equal to 243. Hence, (20+8) × 8 = 224, which is less than 243.

Step 4: Now subtract 224 from 243, and we get 19 as the new remainder, and 18 as a quotient.

Step 5: The new quotient obtained is 18, and double that. Hence, we get 36 and assume that 360 is our new divisor. Now, bring down the two zeros and, and we have 1900 as the new dividend.

Step 6: Find the number, such that (360 + new number) × new number should give the product value that should be less than or equal to 1900. Thus, (360+5) × 5 = 1825, which is less than 1900.

Step 7: Subtract 1825 from 1900, and we get 75 as the new reminder.

Step 8: Continue this process until we get the approximate value of the square root of 343 upto three decimal places. (Note: keep the decimal point in the quotient value after bringing down all the values in the dividend).

Step 9: Thus, the approximate value of the square root of 343, √343 is 18.520.

Video Lessons on Square Roots

Visualising square roots

Finding Square roots

Examples

Example 1:

Simplify: (√7√343) + 10

Solution:

Given: (√7√343) + 10 …(1)

We know that the square root of 343 in radical form is 7√7.

Now, substitute √343 = 7√7 in (1) we get,

(√7√343) + 10 = (√7. 7√7)+10

(√7√343) + 10 = 49 + 10

(√7√343) + 10 = 59

Therefore, (√7√343) + 10 = 59

Example 2:

Simplify: 5√343.

Solution:

Given: 5√343.

We know that the square root of 343 in decimal form is 18.520

Therefore, 5√343 = 5(18.520)

5√343 = 92.6

Therefore, 5√343 is 92.60.