QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
\begin{aligned}
& \frac{\sqrt{x^{5}}}{\sqrt[3]{x^{4}}}=x^{\frac{a}{b}} \\
& \text { If } \frac{\sqrt[3]{x^{4}}}{\sqrt[3]{x^{4}}}=x^{\frac{a}{b}} \text { for all positive values of } x \\
& \text { what is the value of } \frac{a}{b} \text { ? }
\end{aligned}
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Step 1:: Identify the numerator and denominator of the given equation.
The numerator is $$\sqrt{x^{5}}$$ and the denominator is $$\sqrt[3]{x^{4}}$$.
Step 2:: Simplify both the numerator and the denominator.
For the denominator, we can rewrite $$\sqrt[3]{x^{4}}$$ as $$x^{\frac{4}{3}}$$.
Step 3:: Set up the equation with the simplified numerator and denominator.
We have $$x^{\frac{5}{2}} = x^{\frac{a}{b}} \cdot x^{\frac{4}{3}}$$.
Step 4:: Use the rule of exponents to combine the base terms on both sides of the equation.
Therefore, we have $$x^{\frac{5}{2}} = x^{\frac{a}{b} + \frac{4}{3}}$$.
The rule of exponents states that when multiplying terms with the same base, we can add the exponents.
Step 5:: Set the exponents equal to each other since the base terms are equal.
We have $$\frac{5}{2} = \frac{a}{b} + \frac{4}{3}$$.
Step 6:: Find a common denominator for the fractions on the right-hand side of the equation.
The common denominator of $$\frac{a}{b}$$ and $$\frac{4}{3}$$ is $$6b$$.
Step 7:: Multiply both the numerator and denominator of each fraction by the necessary factor to obtain the common denominator.
We have $$\frac{5}{2} = \frac{3a + 4b}{6b}$$.
Step 8:: Solve for the ratio \frac{a}{b}.
Since we are looking for the value of the ratio $$\frac{a}{b}$$, we can rewrite this equation as $$\frac{a}{b} = \frac{5}{1} - \frac{4}{3}$$.
Step 9:: Simplify the expression for the ratio.
We can then subtract the fractions to get $$\frac{a}{b} = \frac{11}{3}$$.
We can find a common denominator for the fractions on the right-hand side of the equation.
Final Answer
The value of the ratio \frac{a}{b} is \frac{11}{3}.
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