Problem 33 - Algebra and Trigonometry

Question

\(29-38=\) Simplify the expression. Assume that the letters denote any real numbers. $$ \sqrt[3]{x^{3} y} $$

Step-by-Step Solution

Answer

The simplified form is \( x \sqrt[3]{y} \).

Step 1: Understand the Cube Root

The expression given is \( \sqrt[3]{x^3 y} \). A cube root determines which number, when multiplied by itself three times, gives the original number. So, we want to find a number that when cubed results in \( x^3 y \).

Step 2: Simplify the Cube and Cube Root

The cube root of \( x^3 \) is simply \( x \), because \( (x^3)^{1/3} = x^{3/3} = x^1 = x \). Thus, the part \( \sqrt[3]{x^3} \) simplifies directly to \( x \).

Step 3: Combine Simplified Components

Combine the simplified result from Step 2 and express the simplified form of the entire expression as \( x \cdot \sqrt[3]{y} \). Therefore, \( \sqrt[3]{x^3 y} = x \sqrt[3]{y} \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down an expression to its simplest form. This can make complex equations easier to handle and understand. In this case, we have the cube root expression \( \sqrt[3]{x^3 y} \). By simplifying, we want to express this in the most reduced form possible.
To do this, we separate the components of the expression: \( x^3 \) and \( y \). We know \( x^3 \) is a perfect cube, which simplifies to \( x \) when taken out of the cube root.
For \( y \), which is not a perfect cube in this context, it stays under the cube root sign.
Finally, the expression is simplified to \( x \cdot \sqrt[3]{y} \). This is the cleanest form of the expression where all components possible have been simplified.

Exponent Rules

Exponent rules are essential for simplifying expressions involving powers. These rules help us understand how to manipulate and simplify expressions with exponents effectively.
When dealing with exponents, such as in the expression \( x^3 \), knowing how to manipulate exponents will greatly aid in simplification. The rule used here is the power of a power rule, which states that when you raise a power to another power, you multiply the exponents:

• \( (x^m)^n = x^{m \cdot n} \)

This rule is applied in the step \( (x^3)^{1/3} = x^{3 \cdot (1/3)} = x^1 = x \). Essentially, reducing \( x^3 \) by its cube root involves dividing the exponent (3) by the root (3), resulting in \( x^1 \), or simply \( x \).
Understanding these rules helps in simplifying expressions efficiently, leaving us with simpler terms that are easier to evaluate or further simplify.

Radicals

Radicals often involve understanding roots, like a square root or cube root, within expressions. The cube root, which is noted by the symbol \( \sqrt[3]{...} \), asks what number multiplied by itself three times results in the number inside the radical.
In our expression, \( \sqrt[3]{x^3 y} \), the radical is specifically a cube root. The cube root of \( x^3 \) is straightforward because \( x^3 \) is a perfect cube, producing \( x \). For terms like \( y \) that are not in perfect cube form, they continue to sit under the radical, leading to \( \sqrt[3]{y} \) remaining part of the expression.

• Cube roots simplify compounding by reducing terms to their basic form outside the radical.
• Non-cubed factors stay within the radical as part of the simplified expression.

Having a solid grasp on how radicals operate, especially cube roots in this context, is essential for the effective simplification of expressions that include these elements. This understanding transforms complex radical expressions into simpler, more manageable forms."