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Advanced Math / Equivalent expressions Difficulty: Hard

The expression $6 RootIndex 5 StartRoot 3 Superscript 5 Baseline x Superscript 45 Baseline EndRoot dot RootIndex 8 StartRoot 2 Superscript 8 Baseline x EndRoot$ is equivalent to $a x^{b}$ , where $a$ and $b$ are positive constants and $x greater than 1$ . What is the value of $a + b$ ?

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Explanation

The correct answer is $StartFraction 361 Over 8 EndFraction$ . The rational exponent property is $\sqrt[n]{y^{m}} = y^{\frac{m}{n}}$ , where $y greater than 0$ , $m$ and $n$ are integers, and $n greater than 0$ . This property can be applied to rewrite the given expression $6 RootIndex 5 StartRoot 3 Superscript 5 Baseline x Superscript 45 Baseline EndRoot dot RootIndex 8 StartRoot 2 Superscript 8 Baseline x EndRoot$ as $6 left parenthesis 3 Superscript five fifths Baseline right parenthesis left parenthesis x Superscript StartFraction 45 Over 5 EndFraction Baseline right parenthesis left parenthesis 2 Superscript eight eighths Baseline right parenthesis left parenthesis x Superscript one eighth Baseline right parenthesis$ , or $6 left parenthesis 3 right parenthesis left parenthesis x Superscript 9 Baseline right parenthesis left parenthesis 2 right parenthesis left parenthesis x Superscript one eighth Baseline right parenthesis$ . This expression can be rewritten by multiplying the constants, which gives $36 left parenthesis x Superscript 9 Baseline right parenthesis left parenthesis x Superscript one eighth Baseline right parenthesis$ . The multiplication exponent property is $y^{n} \cdot y^{m} = y^{n + m}$ , where $y greater than 0$ . This property can be applied to rewrite the expression $36 left parenthesis x Superscript 9 Baseline right parenthesis left parenthesis x Superscript one eighth Baseline right parenthesis$ as $36 x Superscript 9 plus one eighth$ , or $36 x Superscript StartFraction 73 Over 8 EndFraction$ . Therefore, $6 \sqrt[5]{3^{5} x^{45}} \cdot \sqrt[8]{2^{8} x} = 36 x^{\frac{73}{8}}$ . It's given that $6 RootIndex 5 StartRoot 3 Superscript 5 Baseline x Superscript 45 Baseline EndRoot dot RootIndex 8 StartRoot 2 Superscript 8 Baseline x EndRoot$ is equivalent to $a x^{b}$ ; therefore, $a equals 36$ and $b = \frac{73}{8}$ . It follows that $a + b = 36 + \frac{73}{8}$ . Finding a common denominator on the right-hand side of this equation gives $a + b = \frac{288}{8} + \frac{73}{8}$ , or $a + b = \frac{361}{8}$ . Note that 361/8, 45.12, and 45.13 are examples of ways to enter a correct answer.