Question

If $c>0$ and $m$ and $n$ are positive integers, which of the following is equivalent to $c^{\frac{m}{n}}$ ? A) $\frac{c^m}{c^n}$ B) $cm-n$ C) $(\sqrt[m]{c}{)}^n$ D) $(\sqrt[n]{c}{)}^m$

Short answer

The short answer is: D) $(\sqrt[n]{c}{)}^m$

Step-by-step solution

Preliminary Observation

$c^{\frac{m}{n}}$ can be rewritten as $\sqrt[n]{c^m}$.

Comparing to Choices

Now let's compare this to the given choices: A) $\frac{c^m}{c^n}$: This simplification is incorrect, as it divides the exponents rather than taking an nth root. B) $cm-n$: This simplification is also incorrect, as it subtracts the two exponents rather than taking an nth root. C) $(\sqrt[m]{c}{)}^n$: This simplification is very close to our observation, but the exponents m and n have been swapped. D) $(\sqrt[n]{c}{)}^m$: This exactly matches our observation that $c^{\frac{m}{n}}=\sqrt[n]{c^m}$. This is the correct answer. So the equivalent expression for $c^{\frac{m}{n}}$ is: D) $(\sqrt[n]{c}{)}^m$

Key concepts

Roots and Radicals

Roots and radicals are essential components of mathematics representing operations that undo exponentiation. When we talk about roots, such as the square root ($\sqrt{c}$) or the nth root ($\sqrt[n]{c}$), we are identifying a number which, when raised to a certain power, results in the original number.
For example, the square root of 9 is 3, because 3 squared ($3^2$) equals 9. Similarly, in the case of $c^{\frac{m}{n}}$, this expression can be represented as the nth root of $c^m$, showing us how exponentiation and roots are inherently connected.
Radicals, like the root symbol, help us work with numbers that result from fractional exponents, further simplifying and solving more complex expressions.

Simplification of Expressions

Simplifying expressions means making them easier to understand or solve, often by breaking them into smaller, more manageable parts. This process is foundational in algebra as it allows us to express equations and functions in their simplest forms, making calculations more straightforward.
In the case of $c^{\frac{m}{n}}$, simplification involves recognizing that the expression can be written as the nth root of $c^m$, or as $\sqrt[n]{c^m}$. This transformation helps in envisioning the expression in a more tangible form that highlights the operation of taking a root.
The goal is to reduce complexity while preserving the core value of the expression, allowing us to better understand and apply mathematical rules and properties.

Positive Integers

Positive integers are whole numbers greater than zero. They include numbers like 1, 2, 3, and so on, but exclude zero and negative numbers. These integers serve as the basic building blocks for more complex mathematical concepts and structures.
When working with powers and roots like $c^{\frac{m}{n}}$, understanding the role of positive integers as exponents and roots is crucial. Here, both $m$ and $n$ are positive integers, meaning they determine the power to which a number is raised and the root that is operated upon.
The concept ensures that the calculations remain within the realm of easily understandable and applicable whole numbers, maintaining simplicity in mathematical analysis and problem-solving.

Exponent Rules

Exponent rules are mathematical guidelines that help us manipulate powers and expressions efficiently. These rules simplify complex operations involving powers, making them easier to understand and compute.
A few basic exponent rules include:

• Product of Powers: $a^m\times a^n=a^{m+n}$
• Quotient of Powers: $\frac{a^m}{a^n}=a^{m-n}$
• Power of a Power: $(a^m{)}^n=a^{m\times n}$
• Zero Exponent: $a^0=1$, for any non-zero $a$

These rules apply also when dealing with fractional exponents, where $c^{\frac{m}{n}}$ utilizes the concept that it corresponds to $\sqrt[n]{c^m}$. Understanding these rules makes it clear why certain expressions are equivalent and allows for proficient manipulation of algebraic expressions.