Question

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt[3]{16 x^{4} y}-3 x \sqrt[3]{2 x y}+5 \sqrt[3]{-2 x y^{4}}$$

Short answer

None of the terms can be combined, so the simplified expression is: \(\root[3]{16x^4y} -3 x \root[3]{2xy} + 5(-x^3 \root[3]{64yx}\root[3] \)

Step-by-step solution

– Simplify each term under the cube-root

Identify and break down the term inside each cube-root. For each term, factorize the number and express the variables to suitable powers.For \(\root[3]{16x^4y} \): \(\root[3]{16} \cdot \root[3]{x^3} \cdot \root[3]{x} \cdot \root[3]{y} \) = 2\(\root[3]{2x^4y} \)For \(\root[3]{2xy} \): \(\root[3]{2xy} \) remains as it is since it is already simplified.For \(\root[3]{-2xy^4} \): \(\root[3]{-2} \cdot \root[3]{xy^3} \cdot \root[3]{y} \) = -\( \root[3]{2x} y \)analyze.

– Simplify the expression

Combine the simplified terms to form the new expression.So, we have:\( 2 \root[3]{2} \root[3]{x} \root[3]{ y}^3 - 3 x \root[3]{2xy} + 5 (- )\root[3]{2 x y^4})\)

- Combine like terms

Combine and simplify the like terms where possible. Since none of these terms are alike, we express it as each term.The expression is:\( 2 \root[3]{2} \root[3]{x^{4}y} - \root[3]{2 x y} + 5* \root[3]{-2 x y^{4}} \)

Key concepts

Cube Roots

Cube roots are a specific type of radical expression where you need to find a number that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). In algebra, we often simplify cube roots to make equations easier to work with. You may see cube roots written as \(\sqrt[3]{...}\), where the small 3 indicates a cube root rather than the more common square root.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. When simplifying radical expressions, your goal is to break down the terms under the root to their simplest form. For instance, to simplify \(\sqrt[3]{16x^4y}\), you need to separate it into smaller parts that can each be easily rooted. You would break it down into \(\sqrt[3]{16} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{x} \cdot \sqrt[3]{y}\), and then simplify each part.
Remember that radical expressions can often be combined or simplified further by using properties of exponents and factoring.

Algebraic Simplification

Algebraic simplification involves breaking down complex expressions into simpler forms. This is often done by factoring numbers and using properties of exponents and radicals. For example, simplifying \(\sqrt[3]{16x^4y}\) requires you to break it down into smaller, manageable parts and then simplify those parts Individually. The steps might include transforming numbers into factors that are easier to take the cube root of and re-indexing powers of variables to an easier-to-understand form.
Simplification helps in solving equations faster and provides a clearer understanding of the components within an equation.

Factoring

Factoring is the process of breaking down numbers or expressions into their component parts, which can then be used to simplify or solve algebraic equations. When dealing with cube roots or square roots, factoring helps to identify parts of the expression that can be simplified. For instance, in the problem given: \(\sqrt[3]{16x^4y}\), 16 can be factored into \(2 \times 2 \times 2 \times 2\) or \(2^4\), allowing us to simplify it further. Similarly, you factor variables by breaking them down into smaller, divisible parts \(e.g., x^4 into x^3 and x\).
Factoring is key to dealing with radical expressions effectively.