Approximate square root of 12 to the nearest tenth

Approximate square root of 12 to the nearest tenth.

The correct answer and explanation is:

Answer: The approximate square root of 12 to the nearest tenth is 3.5.

Explanation:

To find the square root of 12, we want a number which, when multiplied by itself, equals 12.

1. Understanding Square Roots:
   The square root of a number xx is a number yy such that y2=xy^2 = x. Here, we want yy such that y2=12y^2 = 12.
2. Estimating the Square Root:
   First, identify perfect squares near 12:
   • 32=93^2 = 9
   • 42=164^2 = 16
   Since 1212 lies between 99 and 1616, the square root of 12 will lie between 3 and 4.
3. Narrowing Down Between 3 and 4:
   Try numbers between 3 and 4:
   • 3.42=3.4×3.4=11.563.4^2 = 3.4 \times 3.4 = 11.56 (less than 12)
   • 3.52=3.5×3.5=12.253.5^2 = 3.5 \times 3.5 = 12.25 (just above 12)
   Since 3.42<12<3.523.4^2 < 12 < 3.5^2, the square root of 12 lies between 3.4 and 3.5. Because 12 is closer to 12.25 than 11.56, 3.5 is a better estimate.
4. Rounding to the Nearest Tenth:
   The value 3.53.5 is a good approximation of 12\sqrt{12} when rounded to one decimal place.
5. More Precise Calculation (Optional):
   Using a calculator, 12≈3.4641\sqrt{12} \approx 3.4641. Rounded to the nearest tenth, this is 3.53.5.

Summary:

• Square roots of numbers between perfect squares are estimated by checking squares of decimal numbers.
• For 12\sqrt{12}, the value is between 3 and 4 because 32=93^2 = 9 and 42=164^2 = 16.
• Checking 3.423.4^2 and 3.523.5^2 shows the square root is slightly less than 3.5 but rounds to 3.5 when rounded to one decimal.
• Therefore, 12≈3.5\sqrt{12} \approx 3.5 (nearest tenth).