Questions: If the expression (sqrt[3]27 x^4 y^2)/(5 x^(1/3) y) is written in the form a x^b y^c, then what is the product of a, b and c?

Solution Steps

We start with the expression

\[ \frac{\sqrt[3]{27 x^{4} y^{2}}}{5 x^{\frac{1}{3}} y} \]

First, we simplify the numerator:

\[ \sqrt[3]{27 x^{4} y^{2}} = \sqrt[3]{27} \cdot \sqrt[3]{x^{4}} \cdot \sqrt[3]{y^{2}} = 3 \cdot x^{\frac{4}{3}} \cdot y^{\frac{2}{3}} \]

Thus, the expression becomes:

\[ \frac{3 x^{\frac{4}{3}} y^{\frac{2}{3}}}{5 x^{\frac{1}{3}} y} \]

Next, we rewrite the denominator:

\[ 5 x^{\frac{1}{3}} y = 5 x^{\frac{1}{3}} y^{1} \]

Now we can combine the expressions:

\[ \frac{3 x^{\frac{4}{3}} y^{\frac{2}{3}}}{5 x^{\frac{1}{3}} y^{1}} = \frac{3}{5} \cdot \frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}} \cdot \frac{y^{\frac{2}{3}}}{y^{1}} \]

Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we simplify:

\[ \frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}} = x^{\frac{4}{3} - \frac{1}{3}} = x^{\frac{3}{3}} = x^{1} \]

And for \( y \):

\[ \frac{y^{\frac{2}{3}}}{y^{1}} = y^{\frac{2}{3} - 1} = y^{\frac{2}{3} - \frac{3}{3}} = y^{-\frac{1}{3}} \]

Putting it all together, we have:

\[ \frac{3}{5} x^{1} y^{-\frac{1}{3}} = \frac{3}{5} x^{1} \cdot \frac{1}{y^{\frac{1}{3}}} \]

In the form \( a x^{b} y^{c} \), we identify:

• \( a = \frac{3}{5} \)
• \( b = 1 \)
• \( c = -\frac{1}{3} \)

Finally, we calculate the product of \( a, b, \) and \( c \):

\[ a \cdot b \cdot c = \frac{3}{5} \cdot 1 \cdot \left(-\frac{1}{3}\right) = -\frac{3}{15} = -\frac{1}{5} \]

Final Answer