Problem 27 - Prealgebra

Question

Replace each \(\odot\) with \(<,>,\) or \(=\) to make a true statement. $$-\sqrt{18} \odot-4 \frac{3}{8}$$

Step-by-Step Solution

Answer

\(-\sqrt{18} > -4 \frac{3}{8}\).

Step 1: Simplify the Radical Expression

Begin by simplifying \(-\sqrt{18}\). Notice that \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\). So, \(-\sqrt{18} = -3\sqrt{2}\).

Step 2: Convert the Mixed Number to an Improper Fraction

Convert \(-4 \frac{3}{8}\) to an improper fraction. First, multiply the whole number by the denominator: \(4 \times 8 = 32\). Add the numerator: \(32 + 3 = 35\). Since the number is negative, it becomes \(-\frac{35}{8}\).

Step 3: Calculate Approximate Values

Estimate the approximate decimal values of the expressions. \(\sqrt{2} \approx 1.414\), so \(-3\sqrt{2} \approx -3 \times 1.414 = -4.242\). The decimal of \(-\frac{35}{8}\) is \(-4.375\).

Step 4: Compare the Values

Compare \(-4.242\) and \(-4.375\). Since \(-4.242\) is greater than \(-4.375\), replace \(\odot\) with \(>\): \(-3\sqrt{2} > -\frac{35}{8}\).

Step 5: Write the Final Answer

The final expression with the correct comparison operator is \(-\sqrt{18} > -4 \frac{3}{8}\).

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions might sound tricky, but it gets easier once you understand the rules of square roots and factorization. To simplify a radical like \( \sqrt{18} \), we look for its prime factors or perfect square factors within it. First, notice that 18 can be broken down into 9 and 2 because \( 18 = 9 \times 2 \). The number 9 is a perfect square since \( \sqrt{9} = 3 \). Thus, \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \). So, replacing \( \sqrt{18} \) with \( 3\sqrt{2} \) simplifies it to a more manageable expression.
Moreover, remember that if your original expression had a negative sign, like in \( -\sqrt{18} \), it remains outside the simplified radical expression, resulting in \( -3\sqrt{2} \).
By mastering this concept, you can tackle more complex radical expressions confidently in the future!

Converting Mixed Numbers

Converting mixed numbers like \(-4 \frac{3}{8}\) into improper fractions is a useful skill. It makes calculations easier, especially in comparisons or further arithmetic operations. Firstly, look at the whole number (4) and multiply it by the denominator (8): \(4 \times 8 = 32\). Then, add the numerator (3): \(32 + 3 = 35\). Therefore, the improper fraction representing \(4 \frac{3}{8}\) is \(\frac{35}{8}\).
Since our mixed number is negative, ensure that the improper fraction keeps that negative sign, resulting in \(-\frac{35}{8}\).

• Multiplying the whole number by the denominator.
• Adding the numerator.
• Retaining the negative sign.

This streamlined method is relevant in all math operations requiring consistent expression formatting!

Comparing Values

Comparing values involves estimating and contrasting their magnitude effectively. After simplifying and converting each expression, we're left with values like \(-3\sqrt{2}\) and \(-\frac{35}{8}\). To compare, we can estimate the decimal form of each expression.
For \(3\sqrt{2}\), we know \(\sqrt{2} \approx 1.414\), so \(3 \times 1.414 \approx 4.242\). Since it's negative, this becomes \(-4.242\). Similarly, for \(-\frac{35}{8}\), calculate the decimal: divide 35 by 8 to get \(-4.375\).
We see that \(-4.242\) is greater than \(-4.375\) on the number line, meaning as numbers become more negative, they decrease in value. Therefore, replace \(\odot\) with \(>\) in the expression.
This process of converting and comparing helps ensure we make accurate mathematical judgments, especially when faced with negative values.