What the square root of 243
The Correct Answer and Explanation is:
The square root of 243 is approximately 15.588.
Explanation:
To find the square root of 243, you are essentially looking for a number that, when multiplied by itself, equals 243. In mathematical terms, if we let ( x ) represent the square root of 243, then:
[
x^2 = 243
]
Prime Factorization Method:
To simplify square roots, a common method is using prime factorization. This means breaking down 243 into its prime factors.
- Start by dividing 243 by the smallest prime number possible:
- 243 ÷ 3 = 81
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
So, the prime factorization of 243 is:
[
243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5
]
Simplifying the Square Root:
Now that we have ( 243 = 3^5 ), we can apply the rule for square roots. When dealing with square roots, any factor that appears in pairs can be taken out of the square root.
Here, ( 3^5 ) contains two pairs of 3’s:
[
3^5 = 3^2 \times 3^2 \times 3
]
Since ( \sqrt{3^2} = 3 ), we can simplify this as:
[
\sqrt{243} = \sqrt{3^2 \times 3^2 \times 3} = 3 \times 3 \times \sqrt{3} = 9 \sqrt{3}
]
Thus, ( \sqrt{243} = 9 \sqrt{3} ).
Approximation:
Now, let’s estimate ( \sqrt{3} ). The square root of 3 is approximately 1.732, so:
[
9 \times \sqrt{3} \approx 9 \times 1.732 = 15.588
]
Conclusion:
The square root of 243, simplified, is ( 9 \sqrt{3} ), and the decimal approximation is about 15.588. This approach shows how prime factorization helps simplify square roots and gives both an exact and approximate answer.