Problem 74 - Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt[6]{n^{13}} $$

Step-by-Step Solution

Answer

The simplified form is \( n^2 \sqrt[6]{n} \).

Step 1: Understand the Radical Expression

The given expression is \( \sqrt[6]{n^{13}} \). This is a radical expression with an index of 6 and a radicand of \( n^{13} \). Simplifying the radical will require understanding it in terms of exponents.

Step 2: Rewrite the Radicand with Exponents

Recognize that \( \sqrt[6]{n^{13}} \) can be rewritten using fractional exponents as \( (n^{13})^{\frac{1}{6}} \). This is based on the property that \( \sqrt[b]{a} = a^{\frac{1}{b}} \).

Step 3: Apply the Power Rule for Exponents

Use the rule \((a^m)^n = a^{m \cdot n}\), to simplify \( (n^{13})^{\frac{1}{6}} = n^{13 \times \frac{1}{6}} \). This simplifies the exponent as \( n^{\frac{13}{6}} \).

Step 4: Simplify the Fractional Exponent

Divide the fraction \( \frac{13}{6} \) to see how the expression can be further simplified. \( \frac{13}{6} = 2 \frac{1}{6} \) (since 13 divided by 6 goes 2 whole times with a remainder of 1).

Step 5: Break Down the Exponent to Simplify Further

Thus, \( n^{\frac{13}{6}} = n^2 \times n^{\frac{1}{6}} \). This is decomposing the expression based on the mixed number into a whole number and a fractional part.

Key Concepts

Fractional Exponents

Fractional exponents can seem a bit tricky at first, but they are actually a super useful way to represent roots and powers together. These exponents are written as fractions; for example, an exponent of \( \frac{1}{2} \) indicates the square root. Similarly, an exponent of \( \frac{1}{3} \) would represent a cube root. When you see something like \( a^{\frac{m}{n}} \), it generally translates to the \( n \,\text{th} \) root of \( a^m \).

• Numerator \( m \) tells you the power to which the base is raised.
• Denominator \( n \) represents the root you are taking.

In the expression \( \sqrt[6]{n^{13}} \), it is rewritten using fractional exponents as \( n^{\frac{13}{6}} \). This expression combines the ideas of power (raising to 13) and root (taking the sixth root) into one neat package. Understanding this will help you simplify radical expressions more easily.

Power Rule for Exponents

The Power Rule for exponents is a handy tool when dealing with expressions involving powers of powers, or, in other words, nested exponents. The rule states that \((a^m)^n = a^{m \cdot n}\). This means when you raise an exponentiated number to another power, you multiply the exponents together.

In our example, when we simplified \( (n^{13})^{\frac{1}{6}} \), we used this rule to get \( n^{13 \cdot \frac{1}{6}} = n^{\frac{13}{6}} \). By multiplying the exponents, we move from a more complex nested structure to a single, simplified exponent. This makes any further math like simplification or evaluation much smoother. Keeping this rule in mind can save time and reduce errors!

Radical Expressions

Radical expressions involve roots, most commonly represented with a radical symbol (\(\sqrt{}\)). These expressions often show up in algebra, and simplifying them is invaluable for solving equations.

For example, \( \sqrt[6]{n^{13}} \) is a radical expression with a sixth root. Simplifying such an expression involves converting the root into a fractional exponent, in this case \( n^{\frac{13}{6}} \), and then performing any possible simplifications.

• Convert the radical into a fractional exponent for easier manipulation.
• Use the Power Rule to simplify further.

By understanding how radical expressions can be rewritten and simplified, you can tackle more complex algebraic problems with confidence. This also opens the door to solving equations that involve both radical and exponential parts.