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Of the following, which is the closest approximation to \(\frac{32}{\sqrt{0.256}}=\) ?
We need to find the closest approximation to the expression 32 divided by the square root of 0.256. In plain English, we're looking for what number we get when we take 32 and divide it by whatever the square root of 0.256 equals.
The expression we're working with is: \(\frac{32}{\sqrt{0.256}}\)
Process Skill: TRANSLATE - Converting the mathematical expression into a clear understanding of what we need to calculate
Let's make 0.256 easier to work with. When we see a decimal like this, it's helpful to think about what fraction it represents or how we can express it in a simpler form.
0.256 can be written as 256/1000. But there's an even better way to think about this - let's convert it to see if we can find a perfect square:
0.256 = 256/1000 = 256/1000
Now, 256 is actually a perfect square! We know that 16 × 16 = 256, so 256 = 16²
Therefore: 0.256 = 256/1000 = (16²)/(1000)
Now we can find the square root of 0.256 much more easily:
\(\sqrt{0.256} = \sqrt{\frac{256}{1000}} = \sqrt{\frac{16^2}{1000}} = \frac{16}{\sqrt{1000}}\)
To find \(\sqrt{1000}\), let's think about what number when squared gives us something close to 1000:
So \(\sqrt{1000} \approx 32\) (this is a very good approximation)
Therefore: \(\sqrt{0.256} \approx \frac{16}{32} = \frac{1}{2} = 0.5\)
Process Skill: SIMPLIFY - Breaking down the decimal into manageable parts and recognizing perfect squares
Now we can calculate our final answer:
\(\frac{32}{\sqrt{0.256}} \approx \frac{32}{0.5}\)
When we divide by 0.5, that's the same as multiplying by 2:
\(\frac{32}{0.5} = 32 \times 2 = 64\)
Looking at our answer choices:
Our calculated value of 64 is closest to choice C. 60.
The closest approximation to \(\frac{32}{\sqrt{0.256}}\) is 60, which corresponds to answer choice C.
Verification: Our approximation gave us 64, and among the given choices, 60 (choice C) is indeed the closest value to our calculated result.
1. Not recognizing that the decimal 0.256 can be simplified
Many students see 0.256 and immediately try to estimate \(\sqrt{0.256}\) directly without realizing that 0.256 = 256/1000, where 256 is a perfect square (16²). This leads them to make rough guesses instead of using the algebraic simplification that makes the problem much easier.
2. Attempting to calculate the exact decimal value instead of using approximation strategies
Students may try to find the precise decimal value of \(\sqrt{0.256}\) using long division or calculator methods, missing that this is an approximation question where strategic rounding and estimation are more efficient and appropriate.
1. Incorrectly approximating \(\sqrt{1000}\)
When simplifying \(\sqrt{0.256} = \frac{16}{\sqrt{1000}}\), students often struggle with estimating \(\sqrt{1000}\). They might use \(\sqrt{1000} \approx 30\) (since \(30^2 = 900\)) instead of the more accurate \(\sqrt{1000} \approx 32\) (since \(32^2 = 1024\)), leading to \(\sqrt{0.256} \approx \frac{16}{30} \approx 0.53\) instead of the correct \(\frac{16}{32} = 0.5\).
2. Making arithmetic errors when dividing by 0.5
Even after correctly finding \(\sqrt{0.256} \approx 0.5\), students may struggle with the division \(32 \div 0.5\). Some might incorrectly calculate this as \(32 \times 0.5 = 16\) instead of recognizing that dividing by 0.5 is equivalent to multiplying by 2, giving \(32 \times 2 = 64\).
3. Computational errors with fraction simplification
When working with \(\sqrt{\frac{256}{1000}}\), students might make errors in simplifying the fraction or handling the square root of the denominator, such as incorrectly writing \(\sqrt{\frac{256}{1000}} = \frac{\sqrt{256}}{1000}\) instead of \(\frac{\sqrt{256}}{\sqrt{1000}}\).
1. Choosing an answer that's mathematically close but not the closest option
After calculating approximately 64, students might select answer choice B (30) or D (200) without carefully comparing which option is actually closest to their calculated result. They need to recognize that 60 is only 4 units away from 64, making it clearly the best choice.