Calculating The Value Of (-3/8)^2 A Step-by-Step Guide

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The question at hand asks us to evaluate the expression (−38)2{ \left(-\frac{3}{8}\right)^2 }. This involves understanding the concept of squaring a fraction, particularly when it includes a negative sign. Squaring a number means multiplying it by itself. In this case, we are squaring the fraction -3/8. Let's delve into the process step by step to arrive at the correct answer and understand the underlying mathematical principles.

Step-by-Step Solution

To square the fraction -3/8, we multiply it by itself:

(−38)2=(−38)×(−38){ \left(-\frac{3}{8}\right)^2 = \left(-\frac{3}{8}\right) \times \left(-\frac{3}{8}\right) }

When multiplying fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. Also, we need to remember the rule for multiplying negative numbers: a negative number multiplied by a negative number results in a positive number.

So, let's multiply the numerators:

−3×−3=9{ -3 \times -3 = 9 }

And now, let's multiply the denominators:

8×8=64{ 8 \times 8 = 64 }

Combining these results, we get:

(−38)2=964{ \left(-\frac{3}{8}\right)^2 = \frac{9}{64} }

Therefore, the value of (−38)2{ \left(-\frac{3}{8}\right)^2 } is 964{ \frac{9}{64} }.

Why Other Options Are Incorrect

Let's briefly examine why the other options provided are not the correct answer:

  • (A) −964{-\frac{9}{64}}: This option is incorrect because it fails to account for the rule that a negative number squared is positive. The negative signs cancel each other out during multiplication.
  • (C) −616{-\frac{6}{16}}: This option seems to arise from mistakenly multiplying the numerator and denominator by 2 instead of squaring them. It also incorrectly retains a negative sign.
  • (D) 616{\frac{6}{16}}: This option also comes from an incorrect multiplication, multiplying the numerator and denominator by 2. It does not represent the result of squaring the original fraction.

Squaring Negative Fractions: Key Concepts

The Rule of Signs

In mathematics, understanding the rule of signs is crucial, especially when dealing with multiplication and division. The rule states that:

  • A positive number multiplied by a positive number results in a positive number.
  • A negative number multiplied by a negative number results in a positive number.
  • A positive number multiplied by a negative number results in a negative number.
  • A negative number multiplied by a positive number results in a negative number.

This rule is fundamental in determining the sign of the result when performing operations with signed numbers. In the context of squaring a negative fraction, this rule is what ensures that the result is positive. When you multiply a negative fraction by itself, you are essentially multiplying two negative numbers, which, according to the rule, yields a positive result.

Squaring a Fraction

When squaring a fraction, whether it is positive or negative, you are multiplying the fraction by itself. This means that both the numerator and the denominator are multiplied by themselves. For instance, if you have a fraction ab{ \frac{a}{b} } and you want to square it, you perform the following operation:

(ab)2=ab×ab=a×ab×b=a2b2{ \left(\frac{a}{b}\right)^2 = \frac{a}{b} \times \frac{a}{b} = \frac{a \times a}{b \times b} = \frac{a^2}{b^2} }

This process involves squaring both the numerator and the denominator independently. It's a straightforward application of the multiplication of fractions. When the fraction is negative, the same process applies, but you also need to consider the rule of signs, which we discussed earlier.

Common Mistakes to Avoid

  1. Forgetting the Rule of Signs: One of the most common mistakes is to ignore the rule of signs, especially when dealing with negative numbers. Remember that a negative number squared is always positive. This is because the two negative signs cancel each other out during multiplication. Forgetting this rule can lead to incorrect answers.
  2. Incorrectly Squaring the Numerator and Denominator: Another frequent error is squaring only the numerator or only the denominator, or multiplying them by 2 instead of squaring them. It's crucial to square both the numerator and the denominator. Squaring a number means multiplying it by itself, so both parts of the fraction must undergo this operation.
  3. Misunderstanding the Operation: Sometimes, students may misunderstand what squaring a number means. Squaring is not the same as multiplying by 2; it is multiplying the number by itself. This misunderstanding can lead to significant errors in calculations.

Practical Examples

Let's look at some practical examples to solidify these concepts:

  1. Example 1: Squaring a Positive Fraction

    Evaluate (25)2{ \left(\frac{2}{5}\right)^2 }.

    Solution:

    (25)2=25×25=2×25×5=425{ \left(\frac{2}{5}\right)^2 = \frac{2}{5} \times \frac{2}{5} = \frac{2 \times 2}{5 \times 5} = \frac{4}{25} }

  2. Example 2: Squaring a Negative Fraction (Revisited)

    Evaluate (−38)2{ \left(-\frac{3}{8}\right)^2 }.

    Solution:

    (−38)2=(−38)×(−38)=(−3)×(−3)8×8=964{ \left(-\frac{3}{8}\right)^2 = \left(-\frac{3}{8}\right) \times \left(-\frac{3}{8}\right) = \frac{(-3) \times (-3)}{8 \times 8} = \frac{9}{64} }

  3. Example 3: Squaring a Mixed Fraction

    Evaluate (−112)2{ \left(-1\frac{1}{2}\right)^2 }.

    First, convert the mixed fraction to an improper fraction:

    −112=−32{ -1\frac{1}{2} = -\frac{3}{2} }

    Now, square the improper fraction:

    (−32)2=(−32)×(−32)=(−3)×(−3)2×2=94{ \left(-\frac{3}{2}\right)^2 = \left(-\frac{3}{2}\right) \times \left(-\frac{3}{2}\right) = \frac{(-3) \times (-3)}{2 \times 2} = \frac{9}{4} }

    Convert back to a mixed fraction if necessary:

    94=214{ \frac{9}{4} = 2\frac{1}{4} }

Conclusion

In conclusion, the value of (−38)2{ \left(-\frac{3}{8}\right)^2 } is 964{ \frac{9}{64} }. This result is obtained by correctly applying the rules of multiplying fractions and understanding the behavior of negative numbers when squared. Remember, squaring a negative number results in a positive number, and both the numerator and the denominator of a fraction must be squared. By avoiding common mistakes and practicing with various examples, you can confidently tackle similar mathematical problems. Understanding these fundamental concepts is essential for more advanced mathematics, making it a crucial skill to master.

By understanding the rules and practicing with different examples, you can master the skill of squaring fractions, whether they are positive or negative.