Question

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$\left(x^{3} y^{6}\right)^{1 / 3}$$

Short answer

xy^2

Step-by-step solution

Apply the Power Rule

When raising a power to another power, multiply the exponents. The Power Rule states that \( (a^m)^n = a^{mn} \). Apply this rule to each term inside the parenthesis: \( (x^3 y^6)^{1/3} \) becomes \( x^{3 \cdot 1/3} y^{6 \cdot 1/3} \).

Simplify the Exponents

Simplify the exponents by multiplying: \( 3 \cdot 1/3 = 1 \) and \( 6 \cdot 1/3 = 2 \). Therefore, \( x^{3 \cdot 1/3} = x^1 \) and \( y^{6 \cdot 1/3} = y^2 \).

Write the Final Expression

Combine the simplified terms to write the final expression: \( x^1 y^2 \). Since an exponent of 1 is typically not written, the final answer is: \( xy^2 \).

Key concepts

Power Rule

When simplifying exponents, one of the key concepts to understand is the Power Rule. This rule states that when you raise a power to another power, you multiply the exponents. In mathematical terms, the Power Rule can be written as: \( (a^m)^n = a^{mn} \).
Let's look at the exercise provided: \( (x^3 y^6)^{1/3} \).
Here, we have to apply the Power Rule to simplify the expression. You need to multiply each exponent inside the parenthesis by the outer exponent: \( x^{3 * 1/3} \) and \( y^{6 * 1/3} \). This gives you \( x^1 \) and \( y^2 \). Remember, the Power Rule makes complex exponent expressions much simpler.

Multiplying Exponents

Another important concept in simplifying exponents is multiplying exponents. This step usually follows right after applying the Power Rule. When you multiply exponents, make sure to multiply the numerical values of the exponents accurately.
In the given exercise, the exponents inside the parenthesis (3 for x and 6 for y) are each multiplied by the outer exponent (1/3).
To break it down: \( 3 * 1/3 = 1 \) and \( 6 * 1/3 = 2 \).
So \( x^{3 * 1/3} \) simplifies to \( x^1 \), and \( y^{6 * 1/3} \) simplifies to \( y^2 \). Multiplying exponents in this way ensures you reduce complex expressions into simpler, manageable terms.

Positive Exponents

The final concept to touch on is positive exponents. When simplifying or solving expressions involving exponents, it is essential to express your final answer using only positive exponents. This makes the expression neat and standardized.
In our problem, the original exponents were positive, and after applying the Power Rule and simplifying, we ended up with \( x^1 y^2 \). Note that \( x^1 \) can be simply written as \( x \) because any number raised to the power of 1 is the number itself. So, the final simplified expression is \( xy^2 \).
Always make sure your final answer maintains positive exponents for clarity and simplicity.