Problem 44 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{2 x} \cdot \sqrt[3]{4} \cdot \sqrt[3]{2 x^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression (after rationalizing denominators) is \(2.520677x\).

Step 1: Simplify each Cube Root

Each cube root \(\sqrt[3]{2x}\), \(\sqrt[3]{4}\) and \(\sqrt[3]{2x^2}\) can be written as \(2^{1/3}x^{1/3}\), \(2^{2/3}\) and \(2^{2/3}x^{2/3}\) respectively.

Step 2: Combine Similar Terms

We can combine the terms with similar base. The general exponent rule states that \(a^{n}*a^{m}=a^{n+m}\). So, on applying this rule, the expression becomes \(2x\cdot2\cdot2x^2 = 2^{1}x^{1} * 2^{2} * 2^{1}x^{2}=2^{1+2+1}x^{1+2}=2^{4}x^{3}\)

Step 3: Final Simplification

Now, compute the cube root from the simplified expression. The cube root of \(2^{4}\) is \(2^{4/3}=2^{1+1/3}=2.520677\), and the cube root of \(x^{3}\) is \(x\). Hence, the final simplified form of the given expression is \(2.520677x\).

Key Concepts

Rationalization

Rationalization is a process used in algebra to eliminate radicals, such as square roots or cube roots, from a denominator.
It often involves multiplying the numerator and the denominator of a fraction by a conjugate or an equivalent expression that helps clear the radical.
In the case of cube roots, as in our exercise, rationalization is not as direct as squaring a binomial with square roots. Instead, it involves:

• Recognizing that cube roots in the denominator can complicate computation
• Converting cube roots to exponential form, which helps in easily performing multiplication and addition

For example, if there was a straightforward fraction, say, \( \frac{1}{\sqrt[3]{a}} \), and you wanted to rationalize it, you could transform \( \sqrt[3]{a} \) to \( a^{1/3} \) and then manipulate the expression accordingly.
This simplifies the process of dealing with cube roots in various algebraic expressions.

Exponent Rules

Exponent rules are fundamental in algebra and are especially helpful when simplifying expressions with multiple roots or powers.
Key exponent rules you might need are:

• Product of powers: \( a^m \times a^n = a^{m+n} \)
• Power of a power: \( (a^m)^n = a^{m \cdot n} \)
• Power of a product: \( (ab)^n = a^n \cdot b^n \)

In our problem with cube roots, each term like \( \sqrt[3]{2x} \) can be rewritten using exponent notation; in this case, \( 2^{1/3}x^{1/3} \).
By applying the product of powers, similar bases like 2 and \(x\) can be combined using addition of their exponents.
This allows us to convert seemingly complex root expressions into more manageable forms, making it simpler to maneuver and solve.

Simplification

Simplification involves reducing an expression to its simplest form.
This usually means combining like terms or reducing fractions.
When faced with cube roots or any algebraic terms:

• Look for like bases, which can be added or subtracted using exponent rules
• Rewrite complex roots as powers, like substituting \( \sqrt[3]{a} \) with \( a^{1/3} \)

After rewriting and combining, our original expression was simplified to \( 2^4x^3 \).
Taking the cube root of each, we ended up with \( 2.520677x \), making sure all parts have been rationalized or combined down to their simplest form.
Simplifying helps make calculations more straightforward and leads to a cleaner, more comprehensible result.