Problem 51 - Algebra 2

Question

Write each radical using rational exponents. $$ \left(\sqrt[3]{x^{2}+1}\right)^{2} $$

Step-by-Step Solution

Answer

\((x^2 + 1)^{2/3}\)

Step 1: Understand the Given Expression

The given expression is \( \left(\sqrt[3]{x^{2}+1}\right)^{2} \). This is a nested radical expression, where the expression inside the square is a cube root.

Step 2: Rewrite the Cube Root as a Rational Exponent

The expression \( \sqrt[3]{x^2 + 1} \) can be rewritten using rational exponents. The cube root of any expression \( a \) is \( a^{1/3} \). Therefore, \( \sqrt[3]{x^2 + 1} = (x^2 + 1)^{1/3} \).

Step 3: Apply the Power of a Power Rule

We need to find \( \left( (x^2 + 1)^{1/3} \right)^2 \). Using the power of a power rule \( (a^{m})^n = a^{m\cdot n} \), multiply the exponents: \( (x^2 + 1)^{(1/3)\cdot 2} = (x^2 + 1)^{2/3} \).

Step 4: Final Expression with Rational Exponents

The expression \( \left(\sqrt[3]{x^2 + 1}\right)^{2} \) is rewritten as \( (x^2 + 1)^{2/3} \) using rational exponents.

Key Concepts

Nested Radical Expressions

In mathematics, radical expressions involve roots such as square roots or cube roots. A nested radical expression simply means you have one radical inside another. For example, consider a cube root inside a square root.
This concept can appear daunting, but it's just a matter of breaking down each layer into manageable pieces.
To handle a nested radical, the first step is always to simplify the innermost radical.

• In the expression \( \left(\sqrt[3]{x^{2}+1}\right)^{2} \), the cube root \( \sqrt[3]{x^{2}+1} \) is inside the square raised to the 2nd power.
• By treating each radical separately and applying the relevant rules, you can simplify these expressions systematically.

Power of a Power Rule

One of the essential properties of exponents is the power of a power rule. This rule helps simplify expressions that involve exponents raised to another exponent. When you have something like \( (a^{m})^{n} \), the rule states you multiply the exponents together, resulting in \( a^{m \cdot n} \).
This rule is especially helpful for nested radical expressions when rewritten in terms of rational exponents.

• In our expression, once the cube root \( (x^2 + 1)^{1/3} \) is rewritten in rational form, it is then raised to the 2nd power.
• Using the power of a power rule, \( (x^2 + 1)^{1/3} \) raised to the 2nd power becomes \( (x^2 + 1)^{2/3} \).

This simplification rule makes complex expressions more manageable by converting them into a single exponent.

Cube Root as Rational Exponent

A cube root can be a little less intuitive than a square root, but it's fairly simple to express using rational exponents.
The cube root of a number, say \( a \), can be rethought as \( a^{1/3} \). This is because exponentiation with fractions represents roots.

• For example, \( a^{1/2} \) denotes the square root and \( a^{1/3} \) denotes the cube root.
• When you see \( \sqrt[3]{x^2 + 1} \), it's equivalent to \( (x^2 + 1)^{1/3} \).

Using rational exponents makes it easier to apply other algebraic rules, like the power of a power rule, to simplify the expression further.
This concept forms the backbone of transforming radicals, helping to streamline calculations in both algebra and calculus.