Question

Express in exponential form. $$\sqrt[n]{a^{n} b^{3 n}}$$

Short answer

\(ab^{3}\)

Step-by-step solution

Recall Exponential Rule for Roots

Remember that the nth root of a number can be expressed as that number to the power of 1/n, so \[\sqrt[n]{x} = x^{1/n}\].

Apply the Rule to the Expression

Apply the previously mentioned rule to the given expression. In this case, the expression is \(a^{n} b^{3n}\), so applying the rule we get \[\sqrt[n]{a^{n} b^{3n}} = (a^{n} b^{3n})^{1/n}\].

Use the Power of a Power Rule

Use the power of a power rule, which states that \( (x^m)^n = x^{mn}\). So you compute the exponent for \(a\) and \(b\) separately: \[a^{n} b^{3n})^{1/n} = a^{n \cdot 1/n} b^{3n \cdot 1/n} = a^{n/n} b^{3n/n}\].

Simplify the Exponents

Simplify the fractions in the exponents. Since \(n/n = 1\) and \(3n/n = 3\), the expression simplifies further to: \[a^{n/n} b^{3n/n} = a^{1} b^{3} = ab^{3}\].

Key concepts

Exponential Rule for Roots

Understanding the exponential rule for roots opens up a pathway to simplifying complex mathematical expressions involving roots. Imagine you have a root, like the one in the exercise:
\[\begin{equation}\sqrt[n]{a^{n} b^{3 n}}\end{equation}\]To express this in an exponential form, we apply this powerful rule that connects roots with exponents. In essence, the nth root of a number can be rewritten as that number raised to the fractional power of \[\begin{equation}1/n\end{equation}\].
For example, the square root of 4, which is usually written as \[\begin{equation}\sqrt{4}\end{equation}\], is actually \[\begin{equation}4^{1/2}\end{equation}\] because 2 is the root you're looking at. We can leverage this rule for any root to simplify it into a more handleable exponential expression.

Power of a Power Rule

Dealing with exponents often means encountering situations where we have a power raised to another power, like in the lesson's exercise. You might wonder how to simplify such a scenario. This is where the power of a power rule comes into play. It's a straightforward concept that states if you have an exponent raised to another exponent, you multiply those exponents.
Let's break it down: if you've got \[\begin{equation}(x^m)^n\end{equation}\], you multiply m and n together, resulting in \[\begin{equation}x^{mn}\end{equation}\].
This rule is extremely handy when dealing with complex expressions because it simplifies them down to a single exponent. By utilizing this approach in the example given, we are able to rewrite each part of the expression with a clear, singular exponent, making the expression cleaner and more understandable.

Simplifying Exponents

When you're faced with exponents in algebra, your main goal is often to simplify the expression as much as possible. Simplification can involve reducing fractions, removing negative exponents, or combining like terms. For the given exercise, the simplification process involves dealing with fractional exponents.Take the final step from the exercise: you have exponents that look like \[\begin{equation}a^{n/n}\end{equation}\] and \[\begin{equation}b^{3n/n}\end{equation}\]. When simplifying exponents, remember that dividing like terms results in one, thus \[\begin{equation}a^{1}\end{equation}\] remains, and the expression for b simplifies to \[\begin{equation}b^{3}\end{equation}\].
This step is essential for making the expression more concise and accessible. Simplifying exponents properly is vital, as it directly impacts the clarity and correctness of your solutions in algebra.