Problem 81 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \sqrt[3]{x^{3} y^{6}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( xy^2 \).

Step 1: Understanding the Cube Root

The given expression is \( \sqrt[3]{x^3y^6} \), which signifies the cube root of \( x^3y^6 \). The cube root means raising the expression under the root to the power of \( \frac{1}{3} \).

Step 2: Apply the Power of a Power Rule

Using the rule \((a^m)^n = a^{mn}\), apply to each part of the expression. The cube root (\( \frac{1}{3} \) power) acts as follows: \((x^3)^{\frac{1}{3}} \) becomes \( x^{3 \cdot \frac{1}{3}} = x^1 \) and \( (y^6)^{\frac{1}{3}} \) becomes \( y^{6 \cdot \frac{1}{3}} = y^2 \).

Step 3: Combine the Simplified Terms

The simplified expression from Step 2 is \( x^1 \cdot y^2 \). Since \( x^1 \) is the same as \( x \), the expression simplifies to \( xy^2 \).

Step 4: Express with Rational Exponents

The simplified terms are already in the form with rational exponents, so the final expression \( xy^2 \) is correct and simplified.

Key Concepts

Cube Root

In mathematics, the cube root is a special operation that reverses cubing a number. To find the cube root of a number, we need to determine a value that when multiplied by itself twice (i.e., cubed) results in the original number. For example, the cube root of 27 is 3, because

• 3 \(\times\) 3 \(\times\) 3 = 27

The cube root of a product like \( \sqrt[3]{x^3y^6} \) involves finding a number that when cubed gives us the original expression. In terms of exponents, a cube root is the same as raising a number to the power of \( \frac{1}{3} \). This means that:

• \( \sqrt[3]{x^3} \) becomes \( (x^3)^{\frac{1}{3}} \)
• \( \sqrt[3]{y^6} \) becomes \( (y^6)^{\frac{1}{3}} \)

By applying the exponent of \( \frac{1}{3} \), we simplify the expression by reducing the original powers of variables within the root.

Exponent Rules

Exponent rules are crucial for simplifying expressions with powers. A powerful technique in working with exponents is the "Power of a Power" rule. This rule states:\[(a^m)^n = a^{m\cdot n}\]When dealing with cube roots expressed with rational exponents, these rules become quite handy.To simplify \( (x^3)^{\frac{1}{3}} \), multiply the exponents:

• \(x^{3 \times \frac{1}{3}} = x^1\)

Similarly for \( (y^6)^{\frac{1}{3}} \):

• \(y^{6 \times \frac{1}{3}} = y^2\)

These operations simplify the expressions under the cube root, transforming them into more manageable forms as used in algebraic expressions. Understanding and using these rules effectively can greatly simplify complex expressions and make calculations easier.

Simplified Expressions

In the process of transforming a mathematical expression, reaching a simplified form is the goal. Simplified expressions are easier to interpret and often required in mathematics for clarity and further applications.When simplifying an expression like \( x^1 \cdot y^2 \), we reduce it further because \( x^1 \) simplifies to \( x \). Therefore, the expression becomes \( xy^2 \). This form is not only simpler, but also in compliance with rational exponents.

• This means each variable is expressed as a base raised to a power of a rational number.

Working towards simple and clear forms helps in making further algebraic manipulations more intuitive and ensures that any subsequent operations are based on the most efficient versions of those expressions.