Problem 26 - Prealgebra

Question

Estimate each square root to the nearest integer. Do not use a calculator. $$-\sqrt{125}$$

Step-by-Step Solution

Answer

\(-\sqrt{125}\) is approximately -11.

Step 1: Simplify Under the Square Root

First, recognize that estimating the square root involves finding two perfect squares between which the number 125 falls. The perfect square less than 125 is 121 because its square root is 11, and the perfect square more than 125 is 144 because its square root is 12.

Step 2: Determine Approximate Square Root

Since 125 is closer to 121 than it is to 144, we can estimate that \(\sqrt{125}\) is closer to 11 than it is to 12. Therefore, estimate the square root of 125 to be about 11.

Step 3: Consider the Negative Sign

Recall that the original expression was \(-\sqrt{125}\). Since we estimated \(\sqrt{125}\) to be approximately 11, the negative square root \(-\sqrt{125}\) will be about -11.

Key Concepts

Perfect Squares

Understanding perfect squares is foundational when estimating square roots without a calculator. A perfect square is a number that can be expressed as the square of an integer. For example:

• 121, which is \( 11^2 \)
• 144, which is \( 12^2 \)

These numbers form key reference points when estimating the square roots of non-perfect square numbers. To find an estimation, we seek perfect squares that frame our target number. In the original exercise, our number was 125. We determined:

• The perfect square less than 125 is 121
• The perfect square greater than 125 is 144

By identifying these brackets, we can posit that the square root of 125 will be between 11 and 12, the integers based on these perfect squares.
These values set the stage for the next step in estimation.

Approximation Techniques

In mathematics, approximation techniques are useful when an exact answer isn't necessary or when estimates are all that's possible due to constraints, like when not using a calculator. Approximating square roots involves looking for the perfect squares between which the given number lies. We utilize logical analysis by determining which perfect square our number is nearer.For instance, with 125:

• 125 - 121 = 4
• 144 - 125 = 19

It's evident that 125 is closer to 121 than 144. Therefore, \(\sqrt{125} \)is estimated as nearer to 11, making 11 the rounded approximation. This process shows how:

• Identifying closer perfect squares
• Subtracting the number from these perfect squares

guides the estimation process.

Negative Square Roots

When tasked with finding the negative square root of a number, it's important to remember that the square root operation itself yields a positive result by default unless specified otherwise. In the problem given, the additional negative sign outside the square root symbol indicates the need to consider the negative root. Therefore, once we've approximated \(\sqrt{125} \approx 11,\)we simply attach the negative sign to this result.Hence, \(-\sqrt{125} \approx -11. \)The key points to remember include:

• Understanding square roots default as positive unless indicated
• Acknowledging multiplication by -1 results in the negative counterpart

Utilizing these principles helps in handling negative square roots accurately in mathematical problems.