Problem 45 - Algebra 2

Question

Write each expression in simplest form. Assume that all variables are positive. $$\left(\frac{x^{2}}{x^{-11}}\right)^{\frac{1}{3}}$$

Step-by-Step Solution

Answer

\(\sqrt[3]{\frac{x^{2}}{x^{-11}}}\) simplifies to \(x^{4}\sqrt[3]{x}\).

Step 1: Simplify the Terms inside the Parentheses

The first step is to simplify the terms inside the parentheses. The term \(\frac{x^{2}}{x^{-11}}\) can be rewritten, by using the properties of fractions with exponents, as \(x^{2 - (-11)} = x^{13}\)

Step 2: Apply the Fractional Exponent

Next, the task is to apply the exponent of one third to \(x^{13}\). This can be accomplished by multiplying the exponents. This results in \(x^{13/3} = \sqrt[3]{x^{13}}\).

Step 3: Final Simplification

In the final step, the expression \(\sqrt[3]{x^{13}}\) should be simplified further. Given that \(13 = 3*4 + 1\), the expression can be rewritten as \(x^{4}\sqrt[3]{x}\).

Key Concepts

Exponent Rules

When working with expressions involving exponents, it is essential to understand the basic rules that govern them. Exponent rules are like shortcuts that help us manipulate expressions efficiently.
Here are some fundamental rules:

• Product of Powers Rule: When you multiply two powers with the same base, you add their exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Rule: When you divide two powers with the same base, you subtract the exponents, \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents, \((a^m)^n = a^{m \cdot n}\).
• Negative Exponent Rule: A negative exponent means you take the reciprocal of the base, \(a^{-m} = \frac{1}{a^m}\).

In the original exercise, the term \(\frac{x^2}{x^{-11}}\) uses the quotient of powers rule to simplify to \(x^{2 - (-11)} = x^{13}\). This is a great example of how these rules are applied in simplifying expressions.

Fractional Exponents

Fractional exponents can be challenging but are highly useful in simplifying expressions involving roots. A fractional exponent like \(a^{\frac{m}{n}}\) can be interpreted in a couple of ways:

• It represents the \(n\)-th root of \(a^m\).
• Alternatively, it can represent \((\sqrt[n]{a})^m\).

Both interpretations end with the same result. However, it's often easier to start by taking the root and then applying the exponent.
In the exercise, once we simplify to \(x^{13}\), we are given \(x^{13/3}\). This step uses the idea that a fractional exponent represents a root and a power: \(x^{13/3} = \sqrt[3]{x^{13}}\). This understanding is key to making sense of how exponents and roots work together.

Radicals

Radicals are another way to express roots, such as the square root or cube root of a number. When simplifying expressions, converting between radicals and fractional exponents can often be beneficial.

• The \(n\)-th root of \(a\), written as \(\sqrt[n]{a}\), can also be expressed using fractional exponents as \(a^{1/n}\).
• When you see something like \(\sqrt[3]{x^{13}}\), this is simply \(x^{13/3}\) in radical form.

Radicals and fractional exponents are two sides of the same coin and can be interchanged depending on which makes the problem easier to handle. In the final step of our example, \(\sqrt[3]{x^{13}}\) is expressed by breaking it into \(x^4 \sqrt[3]{x}\) to further simplify the expression.

Positive Variables

In algebra, assuming variables are positive is often essential because it simplifies the manipulation of expressions, especially when radicals and even roots are involved. Positive variables ensure there are no ambiguities in terms of taking roots, as negative values can lead to complex numbers when raised to certain powers.

• When variables are positive, expressions involving square roots, cube roots, and other radicals are straightforward since they will always result in real numbers.
• This assumption simplifies the application of exponent and radical rules, ensuring that all outcomes are real and manageable.

The assumption that \(x\) is positive in our exercise allows us confidently to simplify \(\sqrt[3]{x^{13}}\) without concerns over undefined or complex values. This makes the computation easier and the expression cleaner, leading directly to \(x^4 \sqrt[3]{x}\).