Problem 72 - Algebra A Combined Function

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[10]{a^{5} b^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( a^{1/2} b^{1/2} \) or \( \sqrt{a} \sqrt{b} \).

Step 1: Rewrite the Radical Using Rational Exponents

Start by expressing the radical as a power with an exponent. The expression \( \sqrt[10]{a^5b^5} \) can be rewritten using rational exponents as \( (a^5 b^5)^{1/10} \). This is because the 10th root is equivalent to raising the expression to the power of \( \frac{1}{10} \).

Step 2: Apply the Exponent to Each Factor Inside the Parentheses

Use the property \( (xy)^n = x^n y^n \) to apply the rational exponent \( \frac{1}{10} \) to each factor separately: \( (a^5)^{1/10} \) and \( (b^5)^{1/10} \). This gives us the expression \( a^{5/10} b^{5/10} \).

Step 3: Simplify the Exponents

Simplify the exponents by reducing the fractions: \( a^{5/10} \) simplifies to \( a^{1/2} \) and \( b^{5/10} \) simplifies to \( b^{1/2} \). Therefore, the expression becomes \( a^{1/2} b^{1/2} \).

Step 4: Express as Radical Notation if Needed

Finally, write the simplified expression using radical notation. The expression \( a^{1/2} b^{1/2} \) can be rewritten as \( \sqrt{a} \sqrt{b} \), which is the simplified form of the original radical.

Key Concepts

Simplifying Radicals

When you hear about simplifying radicals, think about making complex-looking mathematical expressions easier to handle. A radical essentially involves a root, like a square root or cube root. In our exercise, the expression given is a 10th root, which sounds complicated. But by using rational exponents, the process becomes simplified. The key here is to break down the radical into smaller parts that are easier to work with. Once it's broken down into rational exponents, you can use the properties of exponents to simplify more efficiently.Consider the original radical \( \sqrt[10]{a^5b^5} \). By rewriting this using rational exponents, it looks more like the expressions you're used to handling, allowing for straightforward simplification.

Exponent Properties

Exponents can sometimes seem intimidating, but they have predictable properties that you can use to your advantage. Two key properties are important in this exercise:

• Multiplying Exponents: \((xy)^n = x^n y^n\)
• Simplifying Exponents: If you have \(x^{m/n}\), it can often be simplified by reducing the fraction \(m/n\)

For \( (a^5b^5)^{1/10} \), the multiplication property lets you apply the exponent to each part inside the parentheses, separately impacting both \(a^5\) and \(b^5\). Then, simplifying with fractions like \(5/10\) to \(1/2\) is the next step, making the expression clearer and less bulky.Understanding these properties allows you to transform a daunting expression into one that is easy to interpret and handle.

Radical Notation

Radical notation is the format we often see as a root symbol (\(\sqrt{}\)). It's the traditional way of expressing roots before rational exponents came into the picture.In the context of this exercise, after simplifying, the expression \(a^{1/2}b^{1/2}\) is transformed back into radical notation. This means translating the exponents back into root symbols, resulting in \(\sqrt{a} \sqrt{b}\). This back-and-forth process between rational exponents and radical notation often allows for insights that are not immediately obvious when the notation is more complex. It gives us a tool for seeing the same situation from multiple angles, optimizing our ability to simplify and understand.

Fractional Exponents

Fractional exponents are another way of expressing roots, and they are extremely powerful in simplifying expressions. A fractional exponent, like \(x^{1/n}\), is equivalent to the \(n\)th root of \(x\).In our exercise, the fractional exponent \((a^5b^5)^{1/10}\) is an efficient way to manage roots. Each part of the original expression is given an exponent of \(5/10\) upon distribution. Once reduced to \(1/2\), it showcases the transformation from radicals to exponents and emphasizes their interchangeable nature. Embracing fractional exponents alongside their simplification can make complex radical expressions less intimidating and more approachable.