Simplify (xy)/( cube root of z)

Solution:

Simplify the expression $\frac{xy}{\sqrt[3]{z}}$

Step:1

Rationalize the denominator by multiplying the expression by $\frac{{\left(\sqrt[3]{z}\right)}^2}{{\left(\sqrt[3]{z}\right)}^2}$ to obtain $\frac{xy}{\sqrt[3]{z}}\cdot \frac{{\left(\sqrt[3]{z}\right)}^2}{{\left(\sqrt[3]{z}\right)}^2}$.

Step:2

Simplify the expression by combining the terms in the denominator.

Step:2.1

Multiply the numerator and the denominator to get $\frac{xy{\left(\sqrt[3]{z}\right)}^2}{\sqrt[3]{z}\cdot {\left(\sqrt[3]{z}\right)}^2}$.

Step:2.2

Express $\sqrt[3]{z}$ as $z^{\frac{1}{3}}$ to rewrite the expression as $\frac{xy{\left(\sqrt[3]{z}\right)}^2}{{\left(z^{\frac{1}{3}}\right)}^3}$.

Step:2.3

Apply the exponent rule $a^m{a}^n=a^{m+n}$ to combine the exponents in the denominator.

Step:2.4

Simplify the denominator by adding the exponents to get $\frac{xy{\left(\sqrt[3]{z}\right)}^2}{z^1}$.

Step:2.5

Recognize that ${\left(\sqrt[3]{z}\right)}^3$ is equal to $z$ and simplify the expression to $\frac{xy{\left(\sqrt[3]{z}\right)}^2}{z}$.

Step:3

Finally, rewrite ${\left(\sqrt[3]{z}\right)}^2$ as $\sqrt[3]{z^2}$ to obtain the simplified form $\frac{xy\sqrt[3]{z^2}}{z}$.

Knowledge Notes:

1. Rationalizing the denominator is a technique used to eliminate radicals or imaginary numbers from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a conjugate or an appropriate form of 1, which does not change the value of the expression.

2. The cube root of a number $z$, denoted as $\sqrt[3]{z}$, can also be expressed as $z^{\frac{1}{3}}$.

3. The power rule for exponents states that $a^m{a}^n=a^{m+n}$, which allows us to combine terms with the same base by adding their exponents.

4. When an exponent is raised to another exponent, the exponents are multiplied: ${\left(a^m\right)}^n=a^{m\cdot n}$.

5. Simplifying expressions often involves combining like terms, applying exponent rules, and reducing fractions to their simplest form.