Problem 480 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[5]{\frac{u^{21}}{u^{11}}}\) (b) \(\sqrt[6]{\frac{v^{30}}{v^{12}}}\)

Step-by-Step Solution

Answer

(a) \ u^2\, (b) \ v^3\

Step 1: Simplify the Exponent in the Radicand of Part (a)

Divide the exponents inside the radical. For the expression inside the fifth root \(\frac{u^{21}}{u^{11}}\), use the property \(\frac{a^m}{a^n} = a^{m-n}\) to get \(u^{21-11} = u^{10}\).

Step 2: Simplify the Root for Part (a)

Apply the fifth root to \(u^{10}\). Using the exponent rule \(\sqrt[n]{a^m} = a^{m/n}\), we get \(\sqrt[5]{u^{10}} = u^{10/5} = u^2\). So, \(\sqrt[5]{\frac{u^{21}}{u^{11}}} = u^2\).

Step 3: Simplify the Exponent in the Radicand of Part (b)

Divide the exponents inside the radical. For the expression inside the sixth root \(\frac{v^{30}}{v^{12}}\), use the property \(\frac{a^m}{a^n} = a^{m-n}\) to get \(v^{30-12} = v^{18}\).

Step 4: Simplify the Root for Part (b)

Apply the sixth root to \(v^{18}\). Using the exponent rule \(\sqrt[n]{a^m} = a^{m/n}\), we get \(\sqrt[6]{v^{18}} = v^{18/6} = v^3\). So, \(\sqrt[6]{\frac{v^{30}}{v^{12}}} = v^3\).

Key Concepts

Exponent Rules

Exponent rules are essential when manipulating expressions with exponents. One key rule is the Quotient of Powers Property, which states that when dividing like bases with exponents, you subtract the exponent in the denominator from the exponent in the numerator. For example, \(\frac{a^{m}}{a^{n}} = a^{m-n}\). Another important rule is the Power of a Power Property, which is used when raising an expression with an exponent to another power. This rule states that you multiply the exponents: \(\left( a^m \right)^n = a^{mn}\). These rules are crucial for simplifying expressions involving exponents.

Fractional Exponents

Fractional exponents are another way to represent roots. Instead of writing the radical symbol, you can use a fraction as the exponent. The denominator of the fraction represents the root, and the numerator represents the power. For example, \(a^{m/n}\) is equivalent to \sqrt[n]{a^m}\. This means the expression is the nth root of \(a\) raised to the mth power. Fractional exponents are useful because they can make calculations easier and more straightforward than using traditional radicals.

Simplifying Radicals

Simplifying radicals often involves reducing expressions to their simplest form. First, if the radicand (the expression under the radical) can be expressed as a power, you should simplify it using exponent rules. For instance, in the expression \sqrt[n]{a^m}\, you can rewrite it using fractional exponents as \a^{m/n}\. Simplification steps include:

• Apply the quotient of powers property when dividing like bases.
• Use the property \(\sqrt[n]{a^m} = a^{m/n}\) to simplify roots.

Applying these techniques makes working with radicals much easier. As shown in the original exercise, these methods are applied to expressions within the radicals to simplify them down to more manageable forms.