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Advanced Math / Equivalent expressions Difficulty: Hard

If $the fraction with numerator the square root of x to the fifth power, end root, and denominator the cube root of x to the fourth power, end root, end fraction, equals x raised to the fraction a over b power$ for all positive values of x, what is the value of $the fraction a over b$ ?

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Explanation

The correct answer is $7 over 6$ . The value of $a, over b$ can be found by first rewriting the left-hand side of the given equation as $the fraction with numerator x raised to the five halves power, and denominator x raised to the four thirds power, end fraction$ . Using the properties of exponents, this expression can be rewritten as $x raised to the open parenthesis, five halves, minus four thirds, close parenthesis, power$ . This expression can be rewritten by subtracting the fractions in the exponent, which yields $x raised to the fraction 7 over 6, power$ . Thus, $a, over b$ is $7 over 6$ . Note that 7/6, 1.166, and 1.167 are examples of ways to enter a correct answer.