Problem 50 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt[3]{16 x^{2}}\)

Step-by-Step Solution

Answer

The simplest radical form is \(2x^{1/3}\cdot \sqrt[3]{2x^2}\).

Step 1: Identify the Expression

The expression given to us is \(\sqrt[3]{16x^2}\). Here, we need to express the given cube root in simplest radical form.

Step 2: Factor Inside the Cube Root

We start by factoring the number 16 under the cube root. The number 16 can be expressed as \(2^4\). So, we can rewrite the expression as follows: \(\sqrt[3]{2^4x^2}\).

Step 3: Simplify Using Properties of Exponents

Recall that \(\sqrt[3]{a^b} = a^{b/3}\). This allows us to simplify \(\sqrt[3]{2^4}\) as \(2^{4/3}\). This means \(\sqrt[3]{2^4x^2} = 2^{4/3}x^{2/3}\).

Step 4: Simplify the Radical Expression

The fraction \(4/3\) suggests \(2^{1/3} \times 2\), and \(2/3\) suggests \(x^{1/3} \times x^{1/3}\). Thus, the expression can be further simplified into \(2x^{1/3}\cdot \sqrt[3]{2x^2}\).

Step 5: Write the Final Simplified Form

The expression \(\sqrt[3]{16x^2}\) simplifies to \(2x^{1/3}\cdot \sqrt[3]{2x^2}\). This is the simplest radical form.

Key Concepts

Cube Root

The notion of a cube root can be quite fascinating. In simple terms, the cube root of a given number seeks to find a number that, when multiplied by itself twice (i.e., cubed), yields the original number. For example, the cube root of 27 is 3 because multiplying 3 three times (3 × 3 × 3) results in 27. Expressed mathematically, the cube root of a number appears as \( \sqrt[3]{a} \), where \( a \) is the number in question.
Understanding how to manage the cube root in algebraic expressions like \( \sqrt[3]{16x^2} \) means recognizing how to break down the components inside the root. It’s about finding a simpler equivalent form while adhering to the properties of cube roots.

Properties of Exponents

Exponents are powerful tools in simplifying expressions. The properties of exponents help us to manage expressions involving powers efficiently. Some key properties include:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m\cdot n} \)
• Power of a Quotient: \( (\frac{a}{b})^n = \frac{a^n}{b^n} \)

To simplify \( \sqrt[3]{2^4 x^2} \), we utilize the property that the cube root of a power equates to dividing the exponent by 3. Thus, \( \sqrt[3]{2^4} = 2^{4/3} \), and for \( x^2 \), the result is \( x^{2/3} \). Understanding these properties allows you to efficiently break down and reconstruct expressions.

Factoring

Factoring requires deftly breaking down a number or expression into its basic building blocks or factors that multiply together to give the original entity. Consider the number 16; it can be decomposed into prime factors, specifically \( 2^4 \).
Factoring becomes extremely useful when simplifying expressions under a radical sign, such as cube roots, because it allows us to extract perfect cubes outside of the radical. This action forms a bridge in our solution, simplifying the radical and making it more manageable.

Simplifying Radical Expressions

Simplifying radical expressions involves breaking down the expression to its most reduced form, whereby no further simplification is possible. The objective is to find an equivalent expression that is simpler but retains the same value.
In dealing with an expression like \( \sqrt[3]{16x^2} \), the goal is to pull out any perfect cubes or reduce powers, using properties of exponents and factoring. After factoring and applying the cube root properties, our expression \( \sqrt[3]{16x^2} \) simplifies into \( 2x^{1/3}\cdot \sqrt[3]{2x^2} \).
This approach results in an expression that is much simpler and easier to work with, while maintaining equivalence to the original term.