Convert to Radical Form (27/125)^(-1/3)

Solution:

Step 1:

Invert the fraction to change the negative exponent to a positive one. ${\left(\frac{27}{125}\right)}^{\frac{1}{3}}$

Step 2:

Express the exponent as a cube root using the property $x^{\frac{m}{n}}=\sqrt[n]{x^m}$. $\sqrt[3]{{\left(\frac{27}{125}\right)}^1}$

Step 3:

Recognize that any number raised to the power of $1$ is the number itself. $\sqrt[3]{\frac{27}{125}}$

Step 4:

Present the final expression in its exact radical form and, if necessary, its decimal equivalent.

Exact Form: $\sqrt[3]{\frac{27}{125}}$ Decimal Form: (not provided)

Knowledge Notes:

The problem involves converting an expression with a rational exponent into radical form. The key knowledge points involved in solving this problem include:

1. Negative Exponents: A negative exponent indicates that the base should be taken as the reciprocal and then raised to the positive of the given exponent. For example, $a^{-n}=\frac{1}{a^n}$.

2. Rational Exponents: A rational exponent, such as $\frac{m}{n}$, can be expressed as a root, specifically the $n$-th root of the base raised to the $m$-th power: $x^{\frac{m}{n}}=\sqrt[n]{x^m}$.

3. Cube Root: The cube root of a number $x$, denoted as $\sqrt[3]{x}$, is a value that, when raised to the power of three, gives $x$. It is a special case of the $n$-th root where $n=3$.

4. Simplifying Radicals: When converting to radical form, it's important to simplify the radical expression if possible. In this case, the cube root of the fraction is already in its simplest form.

5. Exact vs. Decimal Form: The exact form of a radical expression is the expression itself, such as $\sqrt[3]{\frac{27}{125}}$. The decimal form is the approximate value of the radical expression when calculated, which is not always possible to express exactly due to the nature of roots.