Problem 56 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt[10]{\frac{w^{15} x^{20}}{y^{5}}} $$

Step-by-Step Solution

Answer

The simplified expression is \( w^{3/2} \cdot x^{2} \cdot y^{-1/2} \).

Step 1: Understand the Radical Expression

The given expression is \( \sqrt[10]{\frac{w^{15} x^{20}}{y^{5}}} \). This is a tenth root of the expression \( \frac{w^{15} x^{20}}{y^{5}} \). The goal is to express this radical in terms of powers with positive exponents.

Step 2: Write the Radical as a Fractional Exponent

A radical \( \sqrt[n]{a} \) is equivalent to \( a^{1/n} \). So, the expression can be rewritten as:\[\left(\frac{w^{15} x^{20}}{y^{5}}\right)^{1/10}\]This converts the tenth root into a power with a fractional exponent.

Step 3: Apply Power to Each Factor in the Fraction

We apply the \( \frac{1}{10} \) exponent to each factor in the fraction separately:\[\left(\frac{w^{15} x^{20}}{y^{5}}\right)^{1/10} = \frac{w^{15 \times (1/10)} \cdot x^{20 \times (1/10)}}{y^{5 \times (1/10)}}\]This step involves distributing the power to both the numerator and the denominator factors.

Step 4: Simplify Exponents

Calculate the new exponents by multiplying:- For \( w\): \( 15 \times \frac{1}{10} = \frac{15}{10} = \frac{3}{2} \)- For \( x\): \( 20 \times \frac{1}{10} = \frac{20}{10} = 2 \)- For \( y\): \( 5 \times \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \)Thus, the expression simplifies to:\[w^{3/2} \cdot x^{2} \cdot y^{-1/2}\]

Key Concepts

Fractional Exponents

In mathematics, exponents are used to denote repeated multiplication. When studying radical expressions, **fractional exponents** provide an easier way to express roots. This is because fractional exponents turn complicated root expressions into simpler forms.

A fractional exponent like \(a^{1/n}\) represents the \(n\)-th root of \(a\). The denominator of the fraction \(n\) indicates the root, while the numerator (which is \(1\) in this case) signifies that it is the first power of that root. For example, \(\sqrt[10]{a}\) can be expressed as \(a^{1/10}\).

When you encounter an expression such as \(\sqrt[10]{\frac{w^{15} x^{20}}{y^{5}}}\), transforming into fractional exponents allows us to use algebraic rules of exponents easily. It simplifies the manipulation and calculations inside the expression because now we can think of the entire expression as raised to a power rather than a root.

Simplifying Radicals

**Simplifying radicals** is about breaking down complicated roots into simpler and more digestible parts. This involves transforming them into expressions with powers and then simplifying these powers into their simplest forms.

Once a radical expression has been converted into fractional exponents, simplifying it involves distributing the root exponent across each term in the expression. For the example \((\frac{w^{15} x^{20}}{y^{5}})^{1/10}\), you distribute the \(\frac{1}{10}\) to each part:

• \(w^{15 \times (1/10)} = w^{3/2}\)
• \(x^{20 \times (1/10)} = x^{2}\)
• \(y^{5 \times (1/10)} = y^{1/2}\)

This process reduces complexity and produces a more manageable form of the expression. While the original may seem intimidating, now we have more straightforward expressions: \(w^{3/2} \cdot x^{2} \cdot y^{-1/2}\). This simpler form is especially helpful for performing further mathematical operations or solving equations.

Exponent Laws

To master radical expressions and fractional exponents, an understanding of **exponent laws** is crucial. These laws regulate how exponents behave under various mathematical operations, providing the foundation to simplify complex equations.

Consider these key exponent laws when dealing with fractional exponents and simplifying radicals:

• Power of a Power: \((a^{m})^n = a^{m \times n}\) enables the simplification of radicals. E.g., \((w^{15})^{1/10} = w^{15 \times 1/10} = w^{3/2}\).
• Quotient of Powers: \(a^m / a^n = a^{m-n}\), useful when distributing powers through divisions.
• Negative Exponent Rule: \(a^{-n} = 1/a^n\) turns negative exponents into reciprocals. This changes \(y^{-1/2}\) into \(1/y^{1/2}\).

These laws work together harmoniously to make complex polynomial divisions, and root extractions much easier. By applying these rules, the expression simplifies seamlessly to a form that's easier to work with in advanced calculations.