Problem 79 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. $$ \sqrt[3]{54 x y^{3}}-y \sqrt[3]{128 x} $$

Step-by-Step Solution

Answer

The simplified expression is \( -y \sqrt[3]{2x}\).

Step 1: Simplify the first term

The first term, \( \sqrt[3]{54 x y^{3}} \) can be broken down into \( \sqrt[3]{2*3^{3} x y^{3}} \), which can simplify to \(3y \sqrt[3]{2x}\) because \( \sqrt[3]{3^3} \) is 3 and \( \sqrt[3]{y^3} \) is y.

Step 2: Simplify the second term

The second term, \( -y \sqrt[3]{128x} \) can be rewritten as \( -y \sqrt[3]{2^{7} x} \). This simplifies to \( -2^2 y \sqrt[3]{2x} \) because the \( \sqrt[3]{2^6} \) is 2\(^2\).

Step 3: Finding Common Terms

The simplified expressions after step 1 and 2 are \(3y \sqrt[3]{2x}\) and \(-2^2 y \sqrt[3]{2x}\). Now, we clearly notice that both have the common expression of \(y \sqrt[3]{2x}\), we can factor that out.

Step 4: Factor common terms

Now we can express the common term \(y \sqrt[3]{2x}\) as once, the expression becomes \( (3-2^2)y \sqrt[3]{2x} \), which further simplifies to \( -y \sqrt[3]{2x}\).

Key Concepts

Simplifying Radicals

Simplifying radicals helps to make expressions more manageable. When dealing with radicals, it's essential to simplify each term before proceeding further. Radicals are expressions that involve roots, like square roots or cube roots. A radical is considered in its simplest form when no perfect powers remain under the root symbol.

To simplify, you break down numbers and variables inside the radical into their factors. For example, with cube roots, you aim to get a perfect cube under the radical to simplify it. In our original exercise, the term \( \sqrt[3]{54 x y^{3}} \) was simplified by identifying \( 54x \ ext{ as } 2 \cdot 3^3 \cdot x \cdot y^3 \). This allowed us to take the cubic roots where possible, resulting in \( 3y \sqrt[3]{2x} \).

Remember, simplifying radicals is all about knowing the powers and breaking them down systematically.

Cube Root

A cube root is the number that, when multiplied by itself three times, gives the original number. It's denoted as \( \sqrt[3]{x} \). Unlike square roots, cube roots can be negative, as multiplying three negative numbers results in a negative number.

In the given problem, you see terms like \( \sqrt[3]{54xy^3} \), where we find the cube root of both constants and variables. For instance, the cube root of \( 3^3 \) is simply 3, and similarly, the cube root of \( y^3 \) is \( y \), which greatly simplifies the expression.

Understanding cube roots helps you recognize perfect cubes quickly and simplify expressions efficiently. Look for factors that can combine to make a perfect cube whenever possible.

Factoring

Factoring involves expressing a number or expression as a product of its factors. When simplifying expressions, factoring is useful to simplify and combine like terms.

In our exercise, the expressions \( 3y \sqrt[3]{2x} \) and \( -2^2 y \sqrt[3]{2x} \) share the common factor \( y \sqrt[3]{2x} \). By factoring this out, we reduced the expression to \( (3 - 2^2) y \sqrt[3]{2x} \). This dramatically simplified the expression to \( -y \sqrt[3]{2x} \).

Whenever you have terms with common factors, factoring them out streamlines computation and makes it easier to solve or simplify an expression.

Polynomial Operations

Polynomial operations involve adding, subtracting, multiplying, or dividing expressions that contain variables raised to whole-number powers. In our exercise, we looked specifically at addition and subtraction.

Our task was to add or subtract terms if possible. After simplifying, we had two expressions with a common factor: \( 3y \sqrt[3]{2x} \) and \( -4y \sqrt[3]{2x} \). By recognizing the common terms, we were able to subtract effectively, resulting in a simpler single term.

• Addition or subtraction of polynomials requires like terms—terms with the same variable and same power.
• Once simplified, combine the coefficients while keeping the variable part unchanged.

Remember, recognizing like terms and combining them appropriately forms the basis for effective polynomial operations.