Problem 63 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt[3]{128 z^{3}}-\sqrt[3]{-16 z^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 6z \cdot \sqrt[3]{2} \).

Step 1: Simplify the First Cubic Root

The expression given is \( \sqrt[3]{128 z^{3}}-\sqrt[3]{-16 z^{3}} \). Start by simplifying \( \sqrt[3]{128 z^{3}} \). Since 128 can be factored as \( 2^7 \), \( \sqrt[3]{128} = 2^{7/3} = 2^2 \cdot 2^{1/3} = 4 \cdot \sqrt[3]{2} \). Next, simplify \( \sqrt[3]{z^3} \), which is \( z \). So, \( \sqrt[3]{128 z^3} = 4z \cdot \sqrt[3]{2} \).

Step 2: Simplify the Second Cubic Root

Simplify the second part of the expression \( \sqrt[3]{-16 z^{3}} \). Notice that \( -16 = -2^4 \), so \( \sqrt[3]{-16} = -2^{4/3} = -2 \cdot \sqrt[3]{2} \). The cubic root of \( z^3 \) is \( z \), so \( \sqrt[3]{-16 z^3} = -2z \cdot \sqrt[3]{2} \).

Step 3: Subtract the Two Simplified Parts

Now, subtract the two simplified expressions \( 4z \cdot \sqrt[3]{2} - (-2z \cdot \sqrt[3]{2}) \). Combine these terms: \((4z + 2z) \cdot \sqrt[3]{2} = 6z \cdot \sqrt[3]{2} \).

Step 4: Final Step: Write the Simplified Expression

The simplified form of the expression is \( 6z \cdot \sqrt[3]{2} \).

Key Concepts

Cubic Roots

Cubic roots are similar to square roots, but instead of finding a number that, when multiplied by itself twice (making it a total of three times), equals the original number, we look for a number that achieves this condition for three multiplications. For example, since \[\sqrt[3]{8} = 2, \]this is because \[2 \times 2 \times 2 = 8.\] Cubic roots come in handy when simplifying expressions involving higher powers. Each time you work with cubic roots, you are effectively "undoing" the cubing of a number, similar to how square roots "undo" squaring. The cube root of a number is represented as \[\sqrt[3]{x},\]where `x` is the number whose cubic root you're calculating. When dealing with negative numbers, like \[\sqrt[3]{-16},\]the cube root can be negative, unlike square roots which remain positive. Thus, \[\sqrt[3]{-16} = -2 \times 2^{1/3}.\]This unique property helps handle various algebraic expressions seamlessly.

Factoring

Factoring is a key algebraic process that involves writing a number or an expression as a product of its factors. This is particularly useful when working with polynomial equations or when trying to simplify expressions involving roots. For example, the number 128 can be factored as a power of 2:

• \(128 = 2^7\)

This means we have expressed 128 in terms of powers of its prime factor, which is 2. Being able to factor numbers and expressions helps a lot when simplifying expressions that involve cubic roots or other powers. It allows us to break down complex numbers into smaller, more manageable parts. We also utilized factoring in the expression \[\sqrt[3]{-16} = -2^4 = -2 \times 2^{1/3},\]helping us simplify the cubed root of a negative number. This step is essential in reducing expressions to their simplest forms.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations such as addition, subtraction, multiplication, and division. In our example \[\sqrt[3]{128 z^{3}}-\sqrt[3]{-16 z^{3}},\]we deal with an algebraic expression that involves a subtraction of two terms. These terms include variables (like \(z\)) and coefficients derived from simplifying the cubic roots. To manage algebraic expressions effectively, you need to identify and isolate similar parts, such as terms that can be rearranged or simplified. In this context, the expression allows us to employ mathematical operations to simplify it into a more straightforward form.

Combining Like Terms

Combining like terms is an algebra technique used to simplify expressions or equations. We do this by identifying terms with the same variable factors and then adding or subtracting these terms. In the solution discussed here, both parts of the expression involve the term \[z \times \sqrt[3]{2}.\]Thus, we can combine these using arithmetic operations. After simplifying the expression \[4z \times \sqrt[3]{2} - (-2z \times \sqrt[3]{2}),\]we can identify the like terms, which are \[4z\ \text{and} -(-2z),\]both accompanied by the cubic root \[\sqrt[3]{2}.\]This addition leads to \[6z \times \sqrt[3]{2},\]audibly demonstrating the rationale for combining like terms. By recognizing and merging these similar terms, we simplify the expression to a new, reduced form.