Problem 59 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(x^{3} y^{2}\right)^{1 / 4}}{\left(x^{-5} y^{-1}\right)^{-1 / 2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{x^{7/4}} \).

Step 1: Apply the Power of a Power Rule

Use the power of a power rule \( (a^m)^n = a^{mn} \) to simplify both the numerator and the denominator. Start by simplifying the numerator: \( \left( x^3 y^2 \right)^{1/4} = x^{3 \cdot \frac{1}{4}} y^{2 \cdot \frac{1}{4}} = x^{\frac{3}{4}} y^{\frac{1}{2}} \).

Step 2: Simplify the Denominator

Now, apply the power of a power rule to the denominator: \(([x^{-5} y^{-1}]^{-1/2}\) becomes \(x^{-5 \cdot (-1/2)} y^{-1 \cdot (-1/2)} = x^{\frac{5}{2}} y^{\frac{1}{2}} \).

Step 3: Write the Simplified Expression

Express the expression with the simplified numerator and denominator as: \(\frac{x^{3/4} y^{1/2}}{x^{5/2} y^{1/2}} \).

Step 4: Simplify by Cancelling Terms

Since \(y^{1/2}\) is in both the numerator and the denominator, it cancels out. This leaves \(\frac{x^{3/4}}{x^{5/2}} \).

Step 5: Apply the Quotient Rule

Use the quotient rule \(\frac{a^m}{a^n} = a^{m-n} \) to combine the exponents of \(x\): \(x^{3/4 - 5/2} = x^{3/4 - 10/4} = x^{-7/4}\).

Step 6: Express with Positive Exponents

Rewrite the expression using positive exponents by moving \(x^{-7/4}\) to the denominator, resulting in: \(\frac{1}{x^{7/4}} \).

Key Concepts

Understanding the Power of a Power Rule

When dealing with exponents, the power of a power rule is a fundamental concept that helps to simplify expressions raised to another power. This rule states that

• If you have a base raised to an exponent, which is then raised to another exponent, you multiply the exponents together.

To illustrate this rule with an example, consider offering a general formula: \((a^m)^n = a^{m\cdot n}\).
In our exercise, we simplify both the numerator and the denominator by applying this rule.
- For the numerator, \((x^3 y^2)^{1/4}\), the exponents inside the parentheses are multiplied by \(1/4\) individually: \(x^{3\cdot 1/4} = x^{3/4}\) and \(y^{2\cdot 1/4} = y^{1/2}\).
- For the denominator, \((x^{-5} y^{-1})^{-1/2}\), similarly, the exponents are multiplied by \(-1/2\): \(x^{-5\cdot (-1/2)} = x^{5/2}\) and \(y^{-1\cdot (-1/2)} = y^{1/2}\).
This rule makes it straightforward to break down each part of the expression, requiring just simple multiplication of exponents.

The Quotient Rule Explained

Simplifying fractions with exponents is where the quotient rule comes into play, allowing us to manage exponents in a division. This rule states:

• For the same base, when dividing, subtract the exponents.

This generally looks like \(\frac{a^m}{a^n} = a^{m-n}\).
In our problem, after cancelling out common terms in both the numerator and the denominator, we are left with \(\frac{x^{3/4}}{x^{5/2}}\).
- Applying the quotient rule, we subtract the exponents of \(x\): \(x^{3/4} - x^{5/2}\).
To do this subtraction correctly, make sure the fractions have a common denominator: \(x^{3/4} - 10/4\), simplifying it results in \(x^{-7/4}\).
The quotient rule simplifies calculations by converting divisions into simpler exponent subtraction, making the expression easier to work with.

Tips for Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form using various rules and properties of exponents. Here are some tips to make simplifying easier:

• Apply exponent rules carefully: Understanding and using rules like power of a power and quotient rule is the key to simplifying effectively.
• Cancel consistently: Look for common terms in both the numerator and the denominator, as they can be cancelled out, helping reduce complexity.
• Convert to positive exponents: Expressions are typically left in simplified form with positive exponents by moving terms accordingly.

In our example, after using both rules, the expression is simplified until we achieved \(x^{-7/4}\).
To ensure positive exponents, the expression is rewritten as \(\frac{1}{x^{7/4}}\), moving \(x^{-7/4}\) from numerator to denominator.
This method not only simplifies the expression but also aligns with standard conventions on expressing results.