Question

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$\frac{\left(x^{2} y\right)^{1 / 3}\left(x y^{2}\right)^{2 / 3}}{x^{2 / 3} y^{2 / 3}}$$

Short answer

\[ x^{2/3} y \]

Step-by-step solution

Apply power of a power rule

Simplify \((x^2 y)^{1/3}\) and \( (x y^2)^{2/3} \) by applying the power of a power rule \[ (a^m)^n = a^{m \times n} \]: \((x^{2} y)^{1/3} = x^{2/3} y^{1/3}\) and \( (x y^{2})^{2/3} = x^{2/3} y^{4/3} \)

Multiply the simplified expressions

Combine \( x^{2/3} y^{1/3} \) and \( x^{2/3} y^{4/3} \) by adding exponents for any bases that are the same. \[ x^{2/3} y^{1/3} \times x^{2/3} y^{4/3} = x^{(2/3 + 2/3)} y^{(1/3 + 4/3)} = x^{4/3} y^{5/3} \]

Simplify the fraction

Simplify the fraction \ \frac{x^{4/3} y^{5/3}}{x^{2/3} y^{2/3}} \ by subtracting the exponents: \[ \frac{x^{4/3}}{x^{2/3}} = x^{(4/3 - 2/3)} = x^{2/3} \ \ \frac{y^{5/3}}{y^{2/3}} = y^{(5/3 - 2/3)} = y^{3/3} = y \] This gives us: \[ x^{2/3} y \]

Key concepts

positive exponents

In algebra, exponents are used to represent repeated multiplication of a number by itself. The exponent (or power) shows how many times the base is multiplied by itself. For example, in the expression \(x^3\), the number 3 is the exponent, and it means that \(x\) is multiplied by itself three times: \(x \times x \times x\).

When simplifying expressions, it's important to keep all exponents positive because they are easier to understand and work with. If you encounter a negative exponent, you can convert it to a positive exponent by taking the reciprocal of the base. This helps in maintaining consistency and avoiding confusion.

For example, \(x^{-2}\) can be rewritten as \(\frac{1}{x^2}\). Always strive to express your final answers with positive exponents.

power of a power rule

The power of a power rule is a fundamental concept when working with exponents. It states that when you raise an exponent to another exponent, you multiply the exponents. The rule can be expressed as \((a^m)^n = a^{m \times n}\).

For instance, in the given exercise: \((x^2 y)^{1/3}\) and \((x y^2)^{2/3}\), you apply the power of a power rule:

• \((x^2 y)^{1/3}\) becomes \(x^{2/3} y^{1/3}\)
• \((x y^2)^{2/3}\) becomes \(x^{2/3} y^{4/3}\)

Each term's exponent is multiplied by the outer exponent. This helps in breaking down complex expressions into simpler forms, making the calculations easier.

Remember, always multiply the exponents when using this rule.

multiplying exponents

When multiplying exponents with the same base, simply add the exponents together. This is known as the product of powers rule: \(a^m \times a^n = a^{m + n}\).

In our equation, after simplifying using the power of a power rule, we have \(x^{2/3} y^{1/3}\) and \(x^{2/3} y^{4/3}\). To combine these expressions:

• For the base \(x\): \(x^{2/3} \times x^{2/3} = x^{(2/3 + 2/3)} = x^{4/3}\)
• For the base \(y\): \(y^{1/3} \times y^{4/3} = y^{(1/3 + 4/3)} = y^{5/3}\)

Adding the exponents allows us to simplify the expression effectively. This method provides clarity and keeps the equation compact.

simplifying fractions with exponents

To simplify fractions with exponents, use the quotient of powers rule: \(\frac{a^m}{a^n} = a^{m - n}\). This means you subtract the exponent in the denominator from the exponent in the numerator.

Let's look at our example: \(\frac{x^{4/3} y^{5/3}}{x^{2/3} y^{2/3}}\). We handle each base separately:

• For the base \(x\): \(\frac{x^{4/3}}{x^{2/3}} = x^{(4/3 - 2/3)} = x^{2/3}\)
• For the base \(y\): \(\frac{y^{5/3}}{y^{2/3}} = y^{(5/3 - 2/3)} = y\)

This step involves straightforward subtraction of exponents. Simplifying fractions with exponents makes the expression more manageable and prepares it for further algebraic manipulation if needed.