Problem 25 - Prealgebra

Question

Replace each \(\odot\) with \(<,>,\) or \(=\) to make a true statement. $$\sqrt{80} \odot 9.2$$

Step-by-Step Solution

Answer

Replace \(\odot\) with \(<\) so \(\sqrt{80} < 9.2\).

Step 1: Estimate the Square Root

Find an estimate for \(\sqrt{80}\). This square root is between \(\sqrt{64}\) and \(\sqrt{81}\), which are 8 and 9 respectively. Since 80 is closer to 81 than 64, \(\sqrt{80}\) is roughly between 8.5 and 9.

Step 2: Compare the Estimate with the Given Number

Since \(\sqrt{80} \approx 8.9\) based on our estimation, we compare this to 9.2. Thus, \(\sqrt{80} < 9.2\).

Step 3: Final Replacement Decision

Based on the comparison, we replace \(\odot\) with \(<\) to form the true statement \(\sqrt{80} < 9.2\).

Key Concepts

Square Root Estimation

Estimating square roots is a useful mathematical skill, especially when you don't have a calculator handy. Square roots help us find which number, when multiplied by itself, gives us the original number. For example, the square root of 64 is 8 because \(8 \times 8 = 64\). But what if the number isn't a perfect square, like 80? We need to estimate.

Here's how you estimate: First, identify two perfect squares between which your number lies. In this case, 80 falls between 64 and 81. That means \(\sqrt{80}\) is between \(\sqrt{64}\) and \(\sqrt{81}\), or between 8 and 9. Since 80 is closer to 81 than it is to 64, you can guess that the square root is closer to 9 than to 8.

A simple tip: Try using decimals for a more accurate estimate, like 8.5 or 8.9. Here, we can estimate \(\sqrt{80} \approx 8.9\). Don't worry about it being perfect; estimation is all about getting close.

Comparison of Numbers

Once you have your square root estimate, the next step is comparing it with another number. Comparison helps us determine which number is larger, smaller, or if they are equal. In this exercise, we compare \(\sqrt{80} \approx 8.9\) to 9.2.

When comparing numbers, it's often useful to use a number line in your mind. Visualize where each number falls. Here, 8.9 would be to the left of 9.2 because it is smaller.

Always remember the rule of thumb: **"When less than the compared number, move left."** This means 8.9 is less than 9.2. And if you see numbers very close together like in this case, ensure your estimate is as precise as possible for an accurate comparison.

Inequality Symbols

Inequality symbols are crucial in expressing the relationship between two values:

• \(<\): Less than
• \(>\): Greater than
• \(=\): Equal to

These symbols help us convey how one number stands in relation to another. In real-life situations, these symbols help make decisions based on size, value, or quantity comparisons.

In our example with \(\sqrt{80} \) and 9.2, since \(\sqrt{80} \approx 8.9\), and 8.9 is less than 9.2, we use the \(<\) symbol. Thus, the expression becomes \(\sqrt{80} < 9.2\).

Using the right inequality symbol is key in mathematics to accurately describe numeric relationships. Practicing with examples helps solidify the concept, so try to create your own comparisons using inequality symbols with numbers you deal with day-to-day.