Problem 70 - Elementary Algebra

Question

Simplify. $$ \sqrt{n^{21}} $$

Step-by-Step Solution

Answer

\sqrt{n^{21}} = n^{21/2}

Step 1: Identify the Exponent

Notice that the exponent inside the square root is 21. Rewrite the expression: \ \ \ \ \( \sqrt{n^{21}} \ \ \)

Step 2: Rewrite Using Fractional Exponents

Express the square root as a fractional exponent: \ \ \ \ \[ \sqrt{n^{21}} = n^{21/2} \ \ \ \]

Step 3: Simplify the Exponent

Simplify the exponent \( \frac{21}{2}\ \), which cannot be reduced further: \ \ \ \ \[ n^{21/2} \ \ \ \ \]

Step 4: Write the Final Answer

Write the final simplified form using the simplified fractional exponent: \ \ \ \ \[ \sqrt{n^{21}} = n^{21/2} \ \ \ \]

Key Concepts

Fractional Exponents

Understanding fractional exponents is essential in algebra. A fractional exponent like \(n^{a/b}\) indicates both an exponent and a root. The numerator \(a\) represents the power, and the denominator \(b\) represents the root. For instance, \(n^{21/2}\) means that \(n\) is raised to the 21st power, then the square root of that result is taken. Fractional exponents make it easier to work with roots and powers in a single step. There are some key rules to remember:

• \(n^{1/b} = \sqrt[b]{n}\) means the \(b\)-th root of \(n\).
• \(n^{a/b} = (n^a)^{1/b} = \sqrt[b]{n^a}\) breaks down complex expressions.

Understanding these basics will simplify a lot of algebraic manipulations.

Exponentiation

Exponentiation is the operation of raising a base number to a power. If \(n\) is the base and \(k\) is the exponent, then the expression \(n^k\) means that \(n\) is multiplied by itself \(k\) times. In the context of fractional exponents, exponentiation becomes a bit more complex but follows the same principles.
For example, both whole numbers and fractional exponents follow these properties:

• Product of Powers: \((n^a) \cdot (n^b) = n^{a+b}\)
• Power of a Power: \((n^a)^b = n^{a \cdot b}\)
• Power of a Product: \((mn)^a = m^a \cdot n^a\)

Knowing these rules will help you simplify many algebraic expressions efficiently.

Simplification Process

Simplifying algebraic expressions is crucial for solving more complex problems. The process often involves:

• Identifying exponents or roots involved in the expression.
• Rewriting expression using fractional exponents when necessary.
• Applying exponentiation rules to combine, reduce, or separate parts of the expression.

In the given exercise, we saw these steps in action.
We identified the exponent \(21\) inside the square root, rewrote the square root as a fractional exponent \((n^{21/2})\), and then simplified the expression.
This resulted in the final simplified form: \(n^{21/2}\).
Practicing various problems will make you more confident and efficient in using these techniques.