Problem 31 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt[3]{128}+3 \sqrt[3]{2} $$

Step-by-Step Solution

Answer

The simplified expression is \( 7 \sqrt[3]{2} \).

Step 1: Identify the Cube Root Components

First, let's break down the expression into its cube root components. We have two terms: \( \sqrt[3]{128} \) and \( 3 \sqrt[3]{2} \).

Step 2: Simplify \( \sqrt[3]{128} \)

The cube root of 128 can be simplified by finding the prime factorization of 128. We have \( 128 = 2^7 \). Thus, \( \sqrt[3]{128} = \sqrt[3]{2^7} = 2^{7/3} \). This simplifies further because \( 7 = 6 + 1 \), and therefore, \( 2^{7/3} = 2^{6/3} \times 2^{1/3} = 2^2 \times \sqrt[3]{2} = 4 \sqrt[3]{2} \).

Step 3: Simplify the Entire Expression

Now we substitute the simplified cube root back into the expression: \( \sqrt[3]{128} + 3 \sqrt[3]{2} \). Replacing \( \sqrt[3]{128} \) gives us: \( 4 \sqrt[3]{2} + 3 \sqrt[3]{2} \).

Step 4: Combine Like Terms

Both terms have \( \sqrt[3]{2} \) as a factor. So, we can add the coefficients: \( 4 \sqrt[3]{2} + 3 \sqrt[3]{2} = (4+3) \sqrt[3]{2} = 7 \sqrt[3]{2} \).

Key Concepts

Cube Roots

Cube roots are a type of mathematical operation used to find a number, which when multiplied by itself three times results in a given value. In simpler terms, if you have a number "x" such that

• \( x^3 = y \), then \( x \) is called the cube root of \( y \).

Cube roots are especially useful when simplifying expressions that involve power of threes. This concept can help break down complex numbers into simpler parts, allowing for easier manipulation and solving of algebraic equations.
In practice, you often simplify cube roots by converting the given number into a product of cubes. Identifying any perfect cubes within the factorization can help reduce the complexity of the expression, just like we can determine in this exercise that \( \sqrt[3]{128} = 4 \sqrt[3]{2} \).
Remember that recognizing powers of integers, like \(2^3 = 8 \), is essential while simplifying cube roots.

Prime Factorization

Prime factorization is a way of expressing a number as a product of its prime numbers, which are numbers greater than 1 that have no divisors other than 1 and themselves. This technique is crucial for simplifying expressions involving roots, especially cube roots.
To perform prime factorization, you repeatedly divide the number by its smallest prime until all factors are prime. In the given exercise, we factor 128 as follows:

• 128 is even, so divide by 2: \(128 \div 2 = 64\).
• 64 is even, divide by 2: \(64 \div 2 = 32\).
• 32 is even, divide by 2: \(32 \div 2 = 16\).
• 16 is even, divide by 2: \(16 \div 2 = 8\).
• 8 is even, divide by 2: \(8 \div 2 = 4\).
• 4 is even, divide by 2: \(4 \div 2 = 2\).
• 2 is a prime number.

Therefore, the prime factorization of 128 is \(2^7\). This provides a solid foundation for further simplification, such as converting it to \( \sqrt[3]{2^7} \), which aids in combining like terms in an expression.

Combining Like Terms

Combining like terms is a fundamental algebraic principle used to simplify expressions, helping to make sense of complex calculations. Like terms are terms within an expression that have the exact same variables raised to the same powers.
When you come across an expression containing like terms, such as \(4 \sqrt[3]{2} + 3 \sqrt[3]{2}\), the coefficients of these terms can be added together as they share the same variable portion, \(\sqrt[3]{2}\).
Here's how it works:

• Identify the common factor between terms: in this case, \(\sqrt[3]{2}\).
• Combine the coefficients of the like terms: add 4 and 3.
• The result is \(7 \sqrt[3]{2}\), which is a more simplified version of the expression.

This process helps in achieving a cleaner and more manageable expression, aiding further operations. Always ensure terms truly share identical variables and exponents when combining them.

Rational Exponents

Rational exponents are an alternative way to express roots and powers using fractions. This method can often simplify calculations, making them great tools for dealing with roots, especially cube roots and square roots.
A rational exponent \(n/m\) represents a power and a root: the base is raised to the power \(n\), and then you take the \(m\)th root of that result. For example, in this exercise, \(128^{1/3}\) is equivalent to finding the cube root of 128.
This expression can be further rewritten as a multiplication of its factors:

• First, find the prime factorization: \(2^7\).
• Rewrite the cube root expression: \( \sqrt[3]{2^7} = 2^{7/3} \).
• Simplify by splitting into powers: \( 2^{6/3} \times 2^{1/3} = 2^2 \times \sqrt[3]{2} \).

Understanding rational exponents enables you to simplify expressions more efficiently, transforming roots and power calculations into multiplication and division operations.