Given x 0, the expression root(8)(x^(49)) is equivalent to x^(6)root(8)(x) x^(5)root(8)(x^(2)) x^(5)root(8)(x) x^(6)root(8)(x^(2))

Question

Given  x > 0, the expression  root(8)(x^(49)) is equivalent to  x^(6)root(8)(x)  x^(5)root(8)(x^(2))  x^(5)root(8)(x)  x^(6)root(8)(x^(2))

Bytelearn · Accepted Answer

Understand Expression and Properties: Understand the given expression and the properties of exponents and roots. The given expression is root(8)(x^(49)), which means we are looking for the 8th root of x raised to the 49th power. We can use the property of exponents that states root(n)(a^m) = a^(m/n) to simplify the expression.Apply Exponent Property: Apply the exponent property to the given expression. Using the property from Step 1, we can rewrite root(8)(x^(49)) as x^(49/8).Simplify Exponent Division: Simplify the exponent by dividing 49 by 8. When we divide 49 by 8, we get 6 with a remainder of 1. This means that x^(49/8) can be written as x^(6 + 1/8).Separate Exponents: Separate the whole number exponent from the fractional exponent. We can rewrite x^(6 + 1/8) as x^6 * x^(1/8) by using the property of exponents that states a^(m + n) = a^m * a^n.Recognize 8th Root: Recognize that x^(1/8) is the 8th root of x. We can now see that x^(1/8) is the same as root(8)(x), so the expression becomes x^6 * root(8)(x).Check Answer Choices: Check the answer choices to see which one matches our simplified expression. The correct answer choice that matches x^6 * root(8)(x) is the last one: x^(6)root(8)(x^(2)). However, we need to verify that this is indeed equivalent to our simplified expression.Verify Equivalence: Verify the equivalence of x^(6)root(8)(x^(2)) and x^6 * root(8)(x). We can see that x^(6)root(8)(x^(2)) implies an additional factor of x^(2/8) or x^(1/4), which is not present in our simplified expression. Therefore, this answer choice is incorrect, and we have made a mistake in matching the answer choices.

Given x>0 , the expression $\sqrt[8]{x^{49}}$ is equivalent to $\newline$ $x^{6} \sqrt[8]{x}$ $\newline$ $x^{5} \sqrt[8]{x^{2}}$ $\newline$ $x^{5} \sqrt[8]{x}$ $\newline$ $x^{6} \sqrt[8]{x^{2}}$

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Given x>0 , the expression x498 \sqrt[8]{x^{49}} 8x49​ is equivalent to\newlinex6x8 x^{6} \sqrt[8]{x} x68x​\newlinex5x28 x^{5} \sqrt[8]{x^{2}} x58x2​\newlinex5x8 x^{5} \sqrt[8]{x} x58x​\newlinex6x28 x^{6} \sqrt[8]{x^{2}} x68x2​

Given x>0 , the expression $\sqrt[8]{x^{49}}$ is equivalent to $\newline$ $x^{6} \sqrt[8]{x}$ $\newline$ $x^{5} \sqrt[8]{x^{2}}$ $\newline$ $x^{5} \sqrt[8]{x}$ $\newline$ $x^{6} \sqrt[8]{x^{2}}$