Problem 13 - Algebra 2

Question

Simplify each expression. $$ \sqrt[6]{27 x^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \sqrt{3} \cdot x^{1/2} \).

Step 1: Rewrite the Expression

To simplify the expression \( \sqrt[6]{27x^3} \), first rewrite it in terms of exponentiation as \( (27x^3)^{1/6} \).

Step 2: Separate Base and Variables

Apply the property \( (ab)^n = a^n b^n \) to separate the constant and the variable. Thus, \( (27x^3)^{1/6} \) becomes \( 27^{1/6} \cdot x^{3/6} \).

Step 3: Simplify the Variable Term

The term \( x^{3/6} \) simplifies to \( x^{1/2} \) because \( \frac{3}{6} = \frac{1}{2} \). So, the expression now becomes \( 27^{1/6} \cdot x^{1/2} \).

Step 4: Simplify the Constant Term

To simplify \( 27^{1/6} \), recognize that \( 27 = 3^3 \). Thus, \( (3^3)^{1/6} = 3^{3/6} = 3^{1/2} \), which simplifies to \( \sqrt{3} \).

Step 5: Write Final Simplified Expression

Combine the simplified constant and variable terms to get the final simplified expression: \( \sqrt{3} \cdot x^{1/2} \).

Key Concepts

Exponentiation

Exponentiation refers to the operation where a number, called the base, is raised to a power, which is an exponent. This operation is fundamental in expressing repeated multiplication. It's written as \( a^n \), where \( a \) is the base and \( n \) is the exponent.

When simplifying expressions involving roots, exponentiation is a handy tool as any root can be expressed as a fractional exponent. For instance, a sixth root, like \( \sqrt[6]{27} \), can be rewritten as \( 27^{1/6} \). This rewriting helps in manipulating expressions more conveniently using the properties of exponents.

Understanding how exponentiation works with fractional exponents is crucial. It highlights the relationship between roots and powers:

• A square root is equivalent to raising to the power of \( \frac{1}{2} \).
• A cube root corresponds to a power of \( \frac{1}{3} \).
• In general, an \( n \)-th root is the same as raising to the power of \( \frac{1}{n} \).

Properties of Exponents

The properties of exponents are rules that simplify complex exponential expressions. They are key to breaking down and solving these expressions efficiently. Some basic properties include:

• \((a^m)(a^n) = a^{m+n}\): When multiplying like bases, add the exponents.
• \((a^m)^n = a^{mn}\): When raising a power to another power, multiply the exponents.
• \((ab)^n = a^n b^n\): A product raised to an exponent can be separated into two individual bases each raised to the same exponent.

In our case, we applied the rule \((ab)^n = a^n b^n\) to separate the constant and the variable, which transformed \((27x^3)^{1/6}\) into \(27^{1/6} \cdot x^{3/6}\). This separation simplifies complex expressions and makes each part easier to handle individually. By understanding and applying these properties, we can transform and simplify expressions much more easily.

Simplifying Radicals

Simplifying radicals involves expressing a radical in its simplest form, ensuring that there are no more square roots, cube roots, etc., left in an expression. Applied to our expression \( \sqrt[6]{27x^3} \), simplification involves both breaking down the radical and reducing any fractional exponents into simpler forms.

Start by addressing any constants separately from the variables. The constant 27, for example, can be rewritten in exponential form as \(3^3\). When considering the sixth root, it converts to \((3^3)^{1/6}\), which can be simplified using exponentiation to \(3^{1/2}\), or \(\sqrt{3}\).

For the variable part, \(x^3\) becomes \(x^{3/6}\), simplified to \(x^{1/2}\) or \(\sqrt{x}\). Ultimately, simplification is about reducing the expression to its most basic components, combining and rewriting those components using the methods outlined. By mastering these steps, one can systematically simplify any radical expression.