Simplify (xy)/( cube root of z)
The question is asking to perform an algebraic operation to present the given expression in a simpler form. Specifically, it requires to express the fraction where the numerator is the product of two variables x and y, and the denominator is the cube root of another variable z, in a more simplified or reduced manner without actually solving for any variable values.
$\frac{x y}{\sqrt[3]{z}}$
Answer
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:
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.
The cube root of a number $z$, denoted as $\sqrt[3]{z}$, can also be expressed as $z^{\frac{1}{3}}$.
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.
When an exponent is raised to another exponent, the exponents are multiplied: $\left(a^{m}\right)^{n} = a^{m \cdot n}$.
Simplifying expressions often involves combining like terms, applying exponent rules, and reducing fractions to their simplest form.
