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Select the expression that is equivalent to (1)/((3x^(2)+3)^((4)/(5))) root(5)((3x^(2)+3)^(4)) root(4)((3x^(2)+3)^(5)) (1)/(root(5)((3x^(2)+3)^(4))) (1)/(root(4)((3x^(2)+3)^(5)))

Select the expression that is equivalent to 1(3x2+3)45 \frac{1}{\left(3 x^{2}+3\right)^{\frac{4}{5}}} \newline(3x2+3)45 \sqrt[5]{\left(3 x^{2}+3\right)^{4}} \newline(3x2+3)54 \sqrt[4]{\left(3 x^{2}+3\right)^{5}} \newline1(3x2+3)45 \frac{1}{\sqrt[5]{\left(3 x^{2}+3\right)^{4}}} \newline1(3x2+3)54 \frac{1}{\sqrt[4]{\left(3 x^{2}+3\right)^{5}}}

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Full solution

Q. Select the expression that is equivalent to 1(3x2+3)45 \frac{1}{\left(3 x^{2}+3\right)^{\frac{4}{5}}} \newline(3x2+3)45 \sqrt[5]{\left(3 x^{2}+3\right)^{4}} \newline(3x2+3)54 \sqrt[4]{\left(3 x^{2}+3\right)^{5}} \newline1(3x2+3)45 \frac{1}{\sqrt[5]{\left(3 x^{2}+3\right)^{4}}} \newline1(3x2+3)54 \frac{1}{\sqrt[4]{\left(3 x^{2}+3\right)^{5}}}
  1. Understand given expression: Understand the given expression.\newlineWe are given the expression (1)/((3x2+3)(4)/(5))(1)/((3x^{2}+3)^{(4)/(5)}) and we need to find an equivalent expression among the options provided.
  2. Analyze the options: Analyze the options.\newlineThe given expression is a fraction with a denominator that is a power with a rational exponent. We need to find an option that represents the same relationship.
  3. Compare with first option: Compare the given expression with the first option.\newlineThe first option is (3x2+3)45\sqrt[5]{(3x^{2}+3)^{4}}. This is not in a fractional form, so it cannot be equivalent to the given expression which has a denominator.
  4. Compare with second option: Compare the given expression with the second option.\newlineThe second option is (3x2+3)54\sqrt[4]{(3x^{2}+3)^{5}}. This is also not in a fractional form, so it cannot be equivalent to the given expression which has a denominator.
  5. Compare with third option: Compare the given expression with the third option.\newlineThe third option is (1)/((3x2+3)45)(1)/(\sqrt[5]{(3x^{2}+3)^{4}}). This option has the same base and exponent form as the given expression, and it is also in a fractional form. This suggests that it might be equivalent to the given expression.
  6. Confirm equivalence of third option: Confirm the equivalence of the third option.\newlineThe third option can be rewritten as (1)/((3x2+3)(4/5))(1)/((3x^{2}+3)^{(4/5)}), which is exactly the same as the given expression. Therefore, the third option is equivalent to the given expression.

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