Given x 0, the expression root(3)(x^(14)) is equivalent to x^(4)root(3)(x^(2)) x^(5)root(3)(x^(2)) x^(4)root(3)(x^(3)) x^(5)root(3)(x^(3))

Question

Given  x > 0, the expression  root(3)(x^(14)) is equivalent to  x^(4)root(3)(x^(2))  x^(5)root(3)(x^(2))  x^(4)root(3)(x^(3))  x^(5)root(3)(x^(3))

Bytelearn · Accepted Answer

Understand Given Expression: Understand the given expression and the properties of exponents. The given expression is the cube root of x to the 14th power, which is written as root(3)(x^(14)). We can use the property of exponents that states (x^(a/b)) = root(b)(x^a) to rewrite the expression.Break Down Exponent: Break down the exponent 14 into a multiple of 3 plus a remainder. Since we are dealing with a cube root, we want to express 14 as 3n + r, where n is an integer and r is the remainder when 14 is divided by 3. 14 = 3 * 4 + 2, so n = 4 and r = 2.Rewrite Using Exponent Rule: Rewrite the expression using the exponent rule. Using the result from Step 2, we can rewrite x^(14) as x^(3*4+2), which is the same as (x^(3*4)) * (x^2).Apply Cube Root: Apply the cube root to both parts of the expression. We can now take the cube root of both parts separately, which gives us root(3)(x^(3*4)) * root(3)(x^2).Simplify Exponent: Simplify the cube root of x^(3*4). Since the cube root and the exponent of 3*4 are inverse operations, they cancel each other out, leaving us with x^4.Combine Simplified Parts: Combine the simplified parts of the expression. We now have x^4 * root(3)(x^2), which is the simplified form of the original expression.Check Final Expression: Check the final expression against the given options. The final expression we have is x^4 * root(3)(x^2), which matches one of the given options.

Given x>0 , the expression $\sqrt[3]{x^{14}}$ is equivalent to $\newline$ $x^{4} \sqrt[3]{x^{2}}$ $\newline$ $x^{5} \sqrt[3]{x^{2}}$ $\newline$ $x^{4} \sqrt[3]{x^{3}}$ $\newline$ $x^{5} \sqrt[3]{x^{3}}$

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Given x>0 , the expression x143 \sqrt[3]{x^{14}} 3x14​ is equivalent to\newlinex4x23 x^{4} \sqrt[3]{x^{2}} x43x2​\newlinex5x23 x^{5} \sqrt[3]{x^{2}} x53x2​\newlinex4x33 x^{4} \sqrt[3]{x^{3}} x43x3​\newlinex5x33 x^{5} \sqrt[3]{x^{3}} x53x3​

Given x>0 , the expression $\sqrt[3]{x^{14}}$ is equivalent to $\newline$ $x^{4} \sqrt[3]{x^{2}}$ $\newline$ $x^{5} \sqrt[3]{x^{2}}$ $\newline$ $x^{4} \sqrt[3]{x^{3}}$ $\newline$ $x^{5} \sqrt[3]{x^{3}}$