Given x 0, the expression root(5)(x^(32)) is equivalent to x^(6)root(5)(x^(2)) x^(7)root(5)(x^(2)) x^(7) x^(6)

Question

Given  x > 0, the expression  root(5)(x^(32)) is equivalent to  x^(6)root(5)(x^(2))  x^(7)root(5)(x^(2))  x^(7)  x^(6)

Bytelearn · Accepted Answer

Understand Expression and Properties: Understand the given expression and the properties of exponents. The given expression is root(5)(x^(32)), which means we are looking for the fifth root of x raised to the 32nd power. According to the properties of exponents, taking the nth root of a number is the same as raising that number to the power of 1/n.Apply Exponent Property: Apply the property of exponents to rewrite the expression. We can rewrite the fifth root of x^(32) as x^(32/5). This is because the fifth root is equivalent to raising to the power of 1/5, and when we raise a power to a power, we multiply the exponents.Simplify Exponent Division: Simplify the exponent by dividing 32 by 5. When we divide 32 by 5, we get 6 with a remainder of 2. This means that x^(32/5) can be written as x^(6 + 2/5).Separate Exponents: Separate the whole number exponent from the fractional exponent. We can express x^(6 + 2/5) as the product of x^6 and x^(2/5). This is because of the property of exponents that states that when we have the same base with exponents being added, we can multiply the bases with the respective exponents.Recognize Fifth Root: Recognize that x^(2/5) is the fifth root of x squared. We can rewrite x^(2/5) as root(5)(x^2) because raising to the power of 2/5 is the same as squaring and then taking the fifth root.Combine Final Expression: Combine the results to get the final expression. The final expression is x^6 multiplied by the fifth root of x squared, which is written as x^(6)root(5)(x^(2)).

Given x>0 , the expression $\sqrt[5]{x^{32}}$ is equivalent to $\newline$ $x^{6} \sqrt[5]{x^{2}}$ $\newline$ $x^{7} \sqrt[5]{x^{2}}$ $\newline$ $x^{7}$ $\newline$ $x^{6}$

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Given x>0 , the expression x325 \sqrt[5]{x^{32}} 5x32​ is equivalent to\newlinex6x25 x^{6} \sqrt[5]{x^{2}} x65x2​\newlinex7x25 x^{7} \sqrt[5]{x^{2}} x75x2​\newlinex7 x^{7} x7\newlinex6 x^{6} x6

Given x>0 , the expression $\sqrt[5]{x^{32}}$ is equivalent to $\newline$ $x^{6} \sqrt[5]{x^{2}}$ $\newline$ $x^{7} \sqrt[5]{x^{2}}$ $\newline$ $x^{7}$ $\newline$ $x^{6}$