Problem 16 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt[3]{40 a^{4}} $$

Step-by-Step Solution

Answer

\( 2a \times \sqrt[3]{5a} \)

Step 1: Factorize Inside the Radical

To simplify \( \sqrt[3]{40 a^4} \), start by factorizing the radicand \( 40 a^4 \). Break it down as \( 40 = 2^3 \times 5 \) and the expression becomes \( 2^3 \times 5 \times a^4 \).

Step 2: Apply Cube Root Properties

Using the property \( \sqrt[3]{x^3} = x \), apply the cube root to each part of the factorized expression: \( \sqrt[3]{2^3} = 2 \) and \( \sqrt[3]{a^4} = a \times \sqrt[3]{a} \).

Step 3: Simplify the Expression

Combine the results from the cube root: the cube root of \( 2^3 \) gives 2, while \( 5 \) remains inside the radical. The expression now becomes \( 2a \times \sqrt[3]{5a} \).

Key Concepts

Cube Root

A cube root is a special value that, when multiplied by itself twice (or cubed), gives the original number or expression. If you encounter an expression like \( \sqrt[3]{x} \), it means you are looking for a number that results in \( x \) when multiplied by itself three times.

To break it down simply: \( \sqrt[3]{8} = 2 \) because \( 2 \,\text{cubed equals} \, 8 \). Therefore, finding a cube root is like figuring out what number or expression, if cubed, makes the number inside the radical sign.

Cube roots are useful in simplifying expressions and are particularly important when dealing with volume, as the cube root of a number in cubic units gives a length in linear units.

Factorization

Factorization is the process of breaking down a number or an expression into a product of its factors. These factors, when multiplied together, give you the original number or expression.

For instance, to factorize the number 40, you would break it down to its prime factors: \( 40 = 2^3 \times 5 \). This means that 40 can be expressed as multiplying two raised to the power of three with five.

In terms of simplifying radicals, factorization helps by breaking things down to their basic components, making it easier to identify and separate the perfect cubes (or other perfect powers) to simplify cube roots or other roots effectively.

Radicand

The radicand is the number or expression inside a radical symbol. In the context of cube roots, it's the part you take the cube root of.

Let's say you have \( \sqrt[3]{40a^4} \); here, \( 40a^4 \) is the radicand. When simplifying expressions in their simplest radical form, you're focused on manipulating this radicand by methods like factorization, to extract and divide powers neatly.

Terms within the radicand need careful manipulation: for instance, separating the parts which are perfect cubes (like \( 2^3 \) in \( 40a^4 \)) from those that remain in the radical after simplification, like \( 5a \) here.

Exponentiation

Exponentiation refers to the process of raising a number or an expression to a certain power. For example, if \( x \) is raised to the power of \( n \), it is expressed as \( x^n \).

In the context of simplifying radicals, exponentiation is crucial for identifying which parts of an expression can be simplified when inside a cube root. For example, in \( 40a^4 \), exponentiation is seen in \( a^4 \).

When simplifying \( a^4 \), you see it as \( (a^3) \times a \), meaning \( a^3 \) can be neatly simplified to \( a \) when using cube root properties, leaving \( a \) inside the root as part of the simplest form. Understanding exponentiation in this way makes it easier to break down and simplify complex radical expressions.