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To determine which of the given expressions are equivalent to [tex]\(\frac{750}{512}\)[/tex], let's analyze each one in detail.
1. First expression: [tex]\(\frac{\sqrt[3]{750}}{512}\)[/tex]
- The numerator is the cube root of 750, and the denominator remains 512.
- This is clearly not equivalent to the original expression because you are introducing the cube root operation only to the numerator.
2. Second expression: [tex]\(\frac{5}{8}\)[/tex]
- The second expression intends to simplify [tex]\(\frac{750}{512}\)[/tex]. However, simplifying [tex]\(\frac{750}{512}\)[/tex] does not yield [tex]\(\frac{5}{8}\)[/tex].
- Therefore, this is not equivalent to the original expression.
3. Third expression: [tex]\(\sqrt[3]{\frac{750}{512}}\)[/tex]
- Here, you take the cube root of the fraction [tex]\(\frac{750}{512}\)[/tex].
- Taking the cube root of the fraction [tex]\(\frac{750}{512}\)[/tex] is clearly a different operation compared to keeping the fraction as-is.
- This expression is not equivalent to the original expression.
4. Fourth expression: [tex]\(\frac{\sqrt[3]{750}}{\sqrt[2]{512}}\)[/tex]
- Here, the numerator is the cube root of 750, and the denominator is the square root of 512.
- The introduction of both the cube root in the numerator and the square root in the denominator alters both the numerator and the denominator from the original expression.
- This expression is not equivalent to the original expression.
5. Fifth expression: [tex]\(\frac{5}{8} \sqrt[3]{6}\)[/tex]
- This expression is a product of the fraction [tex]\(\frac{5}{8}\)[/tex] and the cube root of 6.
- This is not equivalent to the original expression since it includes an additional cube root of 6.
6. Sixth expression: [tex]\(\frac{750}{\sqrt[3]{512}}\)[/tex]
- This expression has the denominator changed to the cube root of 512.
- This operation changes the value of the denominator from the original expression, making it not equivalent.
Given these detailed analyses, we can determine that none of the provided expressions are equivalent to the original expression [tex]\(\frac{750}{512}\)[/tex].
So, the correct answer is:
None of the expressions are equivalent to [tex]\(\frac{750}{512}\)[/tex].
1. First expression: [tex]\(\frac{\sqrt[3]{750}}{512}\)[/tex]
- The numerator is the cube root of 750, and the denominator remains 512.
- This is clearly not equivalent to the original expression because you are introducing the cube root operation only to the numerator.
2. Second expression: [tex]\(\frac{5}{8}\)[/tex]
- The second expression intends to simplify [tex]\(\frac{750}{512}\)[/tex]. However, simplifying [tex]\(\frac{750}{512}\)[/tex] does not yield [tex]\(\frac{5}{8}\)[/tex].
- Therefore, this is not equivalent to the original expression.
3. Third expression: [tex]\(\sqrt[3]{\frac{750}{512}}\)[/tex]
- Here, you take the cube root of the fraction [tex]\(\frac{750}{512}\)[/tex].
- Taking the cube root of the fraction [tex]\(\frac{750}{512}\)[/tex] is clearly a different operation compared to keeping the fraction as-is.
- This expression is not equivalent to the original expression.
4. Fourth expression: [tex]\(\frac{\sqrt[3]{750}}{\sqrt[2]{512}}\)[/tex]
- Here, the numerator is the cube root of 750, and the denominator is the square root of 512.
- The introduction of both the cube root in the numerator and the square root in the denominator alters both the numerator and the denominator from the original expression.
- This expression is not equivalent to the original expression.
5. Fifth expression: [tex]\(\frac{5}{8} \sqrt[3]{6}\)[/tex]
- This expression is a product of the fraction [tex]\(\frac{5}{8}\)[/tex] and the cube root of 6.
- This is not equivalent to the original expression since it includes an additional cube root of 6.
6. Sixth expression: [tex]\(\frac{750}{\sqrt[3]{512}}\)[/tex]
- This expression has the denominator changed to the cube root of 512.
- This operation changes the value of the denominator from the original expression, making it not equivalent.
Given these detailed analyses, we can determine that none of the provided expressions are equivalent to the original expression [tex]\(\frac{750}{512}\)[/tex].
So, the correct answer is:
None of the expressions are equivalent to [tex]\(\frac{750}{512}\)[/tex].
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