Problem 507 - Elementary Algebra

Question

In the following exercises, simplify. \(\sqrt[4]{s^{10}}\)

Step-by-Step Solution

Answer

Simply \(\sqrt[4]{s^{10}} = s^{5/2}\).

Step 1: Understand the problem

We need to simplify the expression \(\sqrt[4]{s^{10}}\). This means converting it into a simpler form.

Step 2: Rewrite the radical as an exponent

Remember that a fourth root can be represented as an exponent of 1/4. So, \(\sqrt[4]{s^{10}}\) can be rewritten as \((s^{10})^{1/4}\).

Step 3: Apply the power of a power property

Use the exponent rule \((a^m)^n = a^{m \cdot n}\) to simplify the expression. Here, \(m = 10\) and \(n = 1/4\), so we have: \((s^{10})^{1/4} = s^{10 \cdot (1/4)} = s^{10/4}\).

Step 4: Simplify the exponent

Simplify the exponent by dividing 10 by 4. \(10/4 = 2.5\), so the exponent becomes \(s^{2.5}\). To express it as a fraction, \(2.5 = 5/2\), thus \(s^{2.5} = s^{5/2}\).

Key Concepts

Radicals

Radicals are a way to represent roots of numbers and expressions. A common radical is the square root, symbolized as \(\sqrt{a}\). More generally, the nth root of a number is represented as \(\sqrt[n]{a}\). For example, \(\sqrt[4]{s^{10}}\) is the fourth root of \(s^{10}\). Radicals help simplify expressions and solve equations involving roots. They can also be converted into fractional exponents, which is often useful for simplification.

Exponents

Exponents are a way to express repeated multiplication. For instance, \(a^b\) means that the number \(a\) is multiplied by itself \(b\) times. Exponents are essential in algebra for simplifying expressions and solving equations. In the exercise \(\sqrt[4]{s^{10}}\), \(s^{10}\) represents \(s\) multiplied by itself 10 times. Exponents simplify complex multiplication, making it easier to handle large numbers and expressions.

Power of a Power Property

The power of a power property is a key exponent rule. It states that \((a^m)^n = a^{m \cdot n}\). It simplifies expressions where an exponent is raised to another exponent. For example, in \(\sqrt[4]{s^{10}}\), we rewrite it as \((s^{10})^{1/4}\). By applying the power of a power property, we get \((s^{10})^{1/4} = s^{10 \cdot (1/4)}\), which simplifies to \(s^{10/4}\). This rule helps transform complex exponent expressions into simpler forms.

Fractional Exponents

Fractional exponents are another way to represent roots. Instead of using radicals, you can write roots as exponents. For instance, the fourth root of \(a\) can be written as \(a^{1/4}\). In the problem \(\sqrt[4]{s^{10}}\), rewriting it with fractional exponents gives \((s^{10})^{1/4}\), which simplifies to \(s^{10/4}\) or \(s^{2.5}\). Fractional exponents provide a consistent way to work with roots, exponents, and simplify expressions easily.