Roots and Radicals: TEAS Practice Question with Answer & Explanation

Understanding Square Root Approximations

To find the best decimal approximation for a non-perfect square like $\sqrt{11}$, you must identify the two perfect squares that the number falls between. This is a common requirement for the TEAS 7 Math section, which tests your ability to estimate values without use of a complex calculator.

Step-by-Step Breakdown

1. Identify Perfect Squares: List the perfect squares near the number 11.
   • $3^2 = 9$
   • $4^2 = 16$
2. Bracket the Value: Since 11 is between 9 and 16, we know that $\sqrt{11}$ must be between 3 and 4.
3. Refine the Estimate:
   • Is 11 closer to 9 or 16? It is 2 units away from 9 and 5 units away from 16. Therefore, the answer should be closer to 3 than to 4.
4. Test the Options:
   • 3.1: $3.1 \times 3.1 = 9.61$ (Too low, 1.39 away from 11)
   • 3.3: $3.3 \times 3.3 = 10.89$ (Very close! Only 0.11 away from 11)
   • 3.6: $3.6 \times 3.6 = 12.96$ (Too high, 1.96 away from 11)
   • 3.9: $3.9 \times 3.9 = 15.21$ (Way too high)

Based on these calculations, 3.3 is the most accurate approximation provided in the options.

Why the Other Answers are Incorrect

• A) 3.1: While $3.1^2$ is less than 11, it is much further from 11 than 3.3 is.
• C) 3.6: This value results in almost 13 when squared, making it significantly larger than our target of 11.
• D) 3.9: This is nearly 4. Squaring it gets you very close to 16, which is far from 11.

TEAS Math Tip: The "Squaring" Shortcut

On the TEAS, if you aren't sure how to estimate, quickly square the multiple-choice options. The option whose square is closest to the number inside the radical is your correct answer. Multiplying $3.3 \times 3.3$ takes less than 30 seconds and guarantees accuracy! Bromide: If the number ends in .5 (like 3.5), its square will always end in .25 ($3.5^2 = 12.25$). Use that as a benchmark.