Problem 503 - Elementary Algebra

Question

In the following exercises, simplify. \(\sqrt[12]{b^{12}}\)

Step-by-Step Solution

Answer

The simplified form is b.

Step 1: Identify the Radical and Exponent

The expression to simplify is \(\root{12}{b^{12}} \). Identify that the base is \ b \ and both the index of the radical and the exponent are \ 12 \.

Step 2: Apply the Property of Radicals and Exponents

Use the property \(\root{n}{a^{n}} = a \) when \ a \ is a real number and \ n \ is a positive integer. Here, \ n = 12 \ and \ a = b\.

Step 3: Simplify the Expression

Using the property from the previous step, simplifying \(\root{12}{b^{12}} \) gives \ b \.

Key Concepts

property of radicals

When simplifying expressions involving radicals, it's important to understand the properties that govern radicals. A key property of radicals is \(\root{n}{a^{n}} = a\) when \(a\) is a real number and \(n\) is a positive integer. This means that if you have a radical expression where the index matches the exponent, you can simplify directly to the base.

For example, in the expression \( \root{12}{b^{12}} \), both the index and the exponent are 12. According to the property of radicals, \( \root{12}{b^{12}} = b \). This simplification works because taking the 12th root of \(b^{12}\) 'undoes' the exponentiation by 12, leaving just \(b\).

Knowing and applying this property can make simplifying radical expressions much easier and more intuitive.

exponent rules

Understanding exponent rules is essential for simplifying expressions involving exponents and radicals. One important rule is that when you have an exponent inside a radical, such as \( \root{n}{a^{m}} \), you can rewrite the expression as \( (a^{m})^{1/n} \).

Another crucial rule to remember is that \(a^{m/n} \) can be expressed as \( \root{n}{a^{m}} \). This means that roots and exponents are closely related: a root can be thought of as a fractional exponent.

In the specific problem \( \root{12}{b^{12}} \), we can see these rules in action: \(b^{12/12} = b \). This transformation verifies our simplification and highlights how exponent rules streamline the process.

By understanding these exponent rules, you can effectively manipulate and simplify more complex algebraic expressions that involve both exponents and radicals.

simplifying expressions

Simplifying expressions is fundamentally about making them easier to understand and work with. When you encounter an expression like \( \root{12}{b^{12}} \), your goal is to reduce it to its simplest form.

Here are some steps to follow:

• Identify the base, exponent, and index of the radical.
• Apply the relevant properties of radicals.
• Use exponent rules to ease the simplification.

For our exercise, we recognized the property \( \root{n}{a^{n}} = a \) which allowed us to directly simplify to \(b\). This process not only makes calculations easier but also allows you to see patterns and relationships in algebraic expressions.

Simplifying expressions requires practice and a good grasp of the underlying rules. But once mastered, it makes dealing with complex problems much more manageable.