Problem 32 - Algebra and Trigonometry

Question

\(29-38=\) Simplify the expression. Assume that the letters denote any real numbers. $$ \sqrt[3]{x^{3} y^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \( xy^2 \).

Step 1: Understanding the Problem

We need to simplify the expression \( \sqrt[3]{x^3 y^6} \). This involves understanding what the cube root means and how it affects the powers of the variables within the root.

Step 2: Apply Cube Root Properties

The cube root of a product \( a^n b^m \) can be expressed as the product of the cube roots: \( \sqrt[3]{a^n b^m} = \sqrt[3]{a^n} \cdot \sqrt[3]{b^m} \). We will use this property here.

Step 3: Simplify Cube Root of x

Apply the cube root to \( x^3 \) to get \( \sqrt[3]{x^3} = x^{3/3} = x^1 = x \).

Step 4: Simplify Cube Root of y

Next, apply the cube root to \( y^6 \) to get \( \sqrt[3]{y^6} = y^{6/3} = y^2 \).

Step 5: Combine the Simplified Parts

Combine the results from the simplification steps: \( x \cdot y^2 \). Therefore, the simplified expression is \( xy^2 \).

Key Concepts

Cube Root

The cube root of a number or expression is a value that, when multiplied by itself three times, gives the original number or expression. It's like asking, "What number cubed equals this value?" This operation is expressed as \( \sqrt[3]{a} \), where "3" indicates the cube root.
For example, if you have \( \sqrt[3]{8} \), you're looking for what number multiplies by itself twice more to give 8. That's 2, because \( 2 \times 2 \times 2 = 8 \). Cube roots can also be applied to algebraic terms like \( \sqrt[3]{x^3} \).
Let's consider \( \sqrt[3]{x^3} \). We want a number that, when cubed, results in \( x^3 \). That's just \( x \), because \( x \times x \times x = x^3 \). Therefore, the cube root of \( x^3 \) is \( x \). In general, if the exponent is divisible by 3, the cube root simplifies to the base raised to the power of the exponent divided by 3.

Algebraic Simplification

Algebraic simplification involves making an algebraic expression easier to read and work with, usually by reducing the expression to its simplest form. This often includes factoring, reducing fractions, and combining like terms.
When simplifying expressions that include cube roots, the process often involves breaking down each part of the expression using mathematical rules. You see this when each variable's exponent inside the cube root is divided by 3, such as simplifying \( \sqrt[3]{x^3 y^6} \) to \( x \cdot y^2 \).
The idea is to collect and combine similar terms and apply the properties of math operations like multiplication. The goal is to transform the expression into one that is less complex, without changing its overall value.
Simplification makes the expressions more manageable, especially when dealing with equations that require further solving.

Properties of Exponents

Understanding the properties of exponents is key to simplifying expressions that involve powers. These properties allow us to manipulate exponents in various ways that adhere to the rules of arithmetic. Some basic and essential properties include:

• Multiplying Powers: \( a^m \times a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Power of a Product: \( (ab)^n = a^n \times b^n \)
• Power of a Fraction: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)

When dealing with cube roots, another important property to remember is the root of powers. Specifically, \( \sqrt[n]{a^m} = a^{m/n} \). For our example \( \sqrt[3]{x^3 y^6} \), you can apply this property:

• For \( x^3 \), it becomes \( x^{3/3} = x^1 = x \)
• For \( y^6 \), it becomes \( y^{6/3} = y^2 \)

These properties help simplify complex expressions into something we can easily work with, ensuring each term's integrity is maintained.