Problem 72 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \left(\frac{-32 x^{10}}{y^{4}}\right)^{\frac{1}{5}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{-2x^2}{y^{4/5}} \).

Step 1: Convert the Expression into a Radical Form

We begin by rewriting the expression using radicals. The expression given is \( \left(\frac{-32 x^{10}}{y^{4}}\right)^{\frac{1}{5}} \). This can be rewritten as the fifth root of the fraction: \( \sqrt[5]{\frac{-32 x^{10}}{y^{4}}} \).

Step 2: Apply Radical to Numerator and Denominator Individually

Use the property that \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \). This allows us to rewrite the expression as \( \frac{\sqrt[5]{-32x^{10}}}{\sqrt[5]{y^{4}}} \).

Step 3: Simplify the Numerator

Address the numerator \( \sqrt[5]{-32x^{10}} \). Since \(-32 = -2^5 \), it simplifies to \(-2\). The expression \( x^{10} \) under a fifth root becomes \( (x^{10})^{1/5} = x^{10 \times \frac{1}{5}} = x^2 \). Thus, the simplified form for the numerator is \(-2x^2\).

Step 4: Simplify the Denominator

Address the denominator \( \sqrt[5]{y^{4}} \). Since there is no perfect fifth power in this expression, it becomes \( y^{4/5} \).

Step 5: Write the Final Simplified Expression

Combining the simplified numerator and denominator, we have the final expression: \( \frac{-2x^2}{y^{4/5}} \). This is the radical expression in simplest form.

Key Concepts

Fractional Exponents

Fractional exponents are a way to express roots using the language of exponents. When you see a fractional exponent like \( a^{m/n} \), it can be interpreted as a combination of power and root. The numerator \( m \) indicates the usual power, while the denominator \( n \) represents the degree of the root.
For instance, the expression \( x^{1/5} \) means "the fifth root of \( x \)," or \( \sqrt[5]{x} \). If you have \( x^{3/5} \), this means you would first take the fifth root of \( x \) and then raise that result to the third power.
Understanding fractional exponents allows for easier manipulation of expressions, especially when simplifying radical expressions. By converting between radical notation and fractional exponents, you can simplify and solve expressions with greater ease.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the radical to its simplest form. The aim is to make calculations easier to understand and use.
To simplify a radical, you:

• Look for perfect powers that can be eliminated outside of the radical. For instance, \( \sqrt[5]{-32x^{10}} \) can be broken down by recognizing that \(-32\) is \((-2)^5\). This lets you take \(-2\) out of the radical.
• Apply properties of exponents to simplify variables under roots. For example, \( x^{10} \) becomes \( x^2 \) since \( x^{10} = (x^5)^2 \) and \( \sqrt[5]{x^{10}} = x^2 \).

By following these steps, you can simplify even complex radical expressions to make them easier to handle.

Rational Expressions

Rational expressions, much like fractions, involve a relationship between two polynomials. They can include variables, coefficients, and exponentiation.
When you work with rational expressions, particularly involving radicals, you'll often apply the property \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \). This principle allows for separate simplification of numerators and denominators, as seen when turning \( \sqrt[5]{\frac{-32x^{10}}{y^{4}}} \) into \( \frac{\sqrt[5]{-32x^{10}}}{\sqrt[5]{y^{4}}} \).
The power of understanding rational expressions lies in being able to simplify them to their simplest form, optimize expressions for calculation, and maintain the balance between all components within the equation. By practicing rational manipulation techniques, you develop essential skills that apply broadly across algebraic contexts.