Problem 59 - Intermediate Algebra

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 simplified expression is \(\frac{1}{x^{7/4}}\).

Step 1: Apply the Power of a Power Rule

Start by simplifying the numerator and denominator separately using the power of a power rule: \((a^m)^n = a^{m imes n}\). For the numerator, \((x^3 y^2)^{1/4}\) becomes \(x^{3/4} y^{1/2}\). For the denominator, \((x^{-5} y^{-1})^{-1/2}\) becomes \(x^{5/2} y^{1/2}\).

Step 2: Simplify the Fraction

Now, simplify the expression by dividing the powers of similar bases: \(\frac{x^{3/4} y^{1/2}}{x^{5/2} y^{1/2}}\). This simplifies to \(x^{3/4 - 5/2} y^{1/2 - 1/2}\).

Step 3: Subtract the Exponents

Subtract the exponents for each base separately to simplify further: For \(x\), the exponent: \(\frac{3}{4} - \frac{5}{2} = \frac{3}{4} - \frac{10}{4} = -\frac{7}{4}\), and for \(y\), the exponent is \(\frac{1}{2} - \frac{1}{2} = 0\).

Step 4: Express with Positive Exponents

Convert any negative exponents to positive by rewriting them on the other side of the fraction: \(x^{-7/4} y^0 = \frac{1}{x^{7/4}}\). Since \(y^0 = 1\), it drops out, and the final simplified expression is \(\frac{1}{x^{7/4}}\).

Key Concepts

Power of a Power Rule

The power of a power rule is crucial when dealing with exponents, particularly in expressions like \((x^m)^n\). This rule states that when an exponent is taken to another exponent, the powers are multiplied, resulting in the new expression: \(x^{m \times n}\). For example, if we have \((x^3 y^2)^{1/4}\), you apply the rule to get \(x^{3/4} y^{1/2}\). The key step here is recognizing that you multiply the original exponents \((3\) for \(x\) and \(2\) for \(y)\), by the outer exponent \(1/4\). This simplifies the expression without changing its value.

• When simplifying nested powers, multiply the exponents.
• It applies to each variable separately in a term.

The same rule is applied to the denominator \((x^{-5} y^{-1})^{-1/2}\), changing it to \(x^{5/2} y^{1/2}\). Notice that the negative in the outer exponent inverts the sign of the exponents of the bases.Understanding this rule lays the foundation for simplifying complex exponent expressions.

Simplifying Expressions

Simplifying expressions is all about making them easier to work with by reducing their complexity. In the case of the given exercise, we first dealt with powers using the power of a power rule and then simplified the entire expression. This involves dividing the similar base terms in the numerator and the denominator.

When you have \(\frac{x^{3/4} y^{1/2}}{x^{5/2} y^{1/2}}\), you simplify by subtracting exponents of similar bases:

• For \(x\): \(\frac{3}{4} - \frac{5}{2} = -\frac{7}{4}\).
• For \(y\): \(\frac{1}{2} - \frac{1}{2} = 0\).

This gives us \(x^{-7/4} y^0\). This simplification helps to clean up the expression, making it easier and more useful for further calculations or analysis. Expressing with \(y^0 = 1\) essentially removes \(y\) from the equation entirely.

Positive Exponents

Expressions with positive exponents tend to be clearer and preferred in their simplified form. Negative exponents might look complex, but they can be easily transformed into positive ones by moving them across the fraction line.

In our example, we had \(x^{-7/4}\) after simplification. To convert this to positive exponents, you shift it to the denominator:

• Start with \(x^{-7/4} y^0\), with \(y^0 = 1\).
• Rewrite as \(\frac{1}{x^{7/4}}\).

This results in an expression that is simple and consists only of positive exponents. This conversion is valuable not just for simplification but also for aligning with mathematical conventions and making expressions more intuitive for further computations or applications.