Problem 37 - Intermediate Algebra

Question

Find each root. Assume that all variables represent nonnegative real numbers. $$ \sqrt[6]{64 x^{12}} $$

Step-by-Step Solution

Answer

The sixth root of \( \sqrt[6]{64 x^{12}} \) is \( 2x^2 \).

Step 1: Understand the radical expression

The expression \( \sqrt[6]{64 x^{12}} \) is a sixth root, which means we are looking for a number that, when raised to the power of 6, equals \( 64x^{12} \). This involves both the numerical and variable parts of the expression.

Step 2: Simplify the numerical part

The number 64 under the radical is a perfect sixth power, because \( 2^6 = 64 \). Therefore, the sixth root of 64 is 2.

Step 3: Simplify the variable part

The variable part is \( x^{12} \). To find the sixth root, divide the exponent by 6: \( x^{12/6} = x^2 \).

Step 4: Combine simplified parts

Combine the results from Steps 2 and 3 to find the sixth root of the entire expression. Thus, \( \sqrt[6]{64x^{12}} = 2x^2 \).

Key Concepts

Understanding the Sixth Root

In mathematics, a sixth root refers to the number that, when multiplied by itself six times, produces the original number under the root. It's part of a broader category of nth roots, corresponding to an exponent of 1/6.
For example, the sixth root of 64 is 2 because when you raise 2 to the power of 6, you obtain 64:

• 2 multiplied by itself six times: \[ 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \]

Similarly, when working with numbers or expressions under a sixth root, look for values that simplify easily. If an expression includes variables with exponents, remember they will also follow the nth root rule, dividing the exponent by 6. This simplifies calculations, especially in problems like simplifying radical expressions.

Exponents and Their Role

Exponents are a fundamental mathematical concept that helps express repeated multiplication compactly. For instance, in the expression \( x^{12} \), 12 is the exponent, indicating that \( x \) is multiplied by itself 12 times.

When we deal with expressions under a radical, exponents play an essential role in simplifying them. Finding the sixth root of a base with an exponent, like \( x^{12} \), involves dividing the exponent by 6:

• \( x^{12/6} = x^2 \)

This division reflects the operation of the root, where the base remains the same while the exponent is a fraction. This way, it becomes much simpler to manage large powers or multiple terms when you simplify radical expressions. Exponents, along with roots, give us powerful means to unravel complex algebraic structures.

Simplifying Variable Expressions

Variable expressions are mathematical phrases involving variables, numbers, and operators. Simplifying these expressions can often involve breaking down parts of the equation using exponent rules and roots. Variable expressions like \( x^{12} \) under a radical symbol require careful handling: you use the root to adjust the power attached to each variable.
For example, with the sixth root, each variable's exponent within the expression should be divided by 6, as shown in the previous parts:

• From \( x^{12} \), we simplify as \( x^{12/6} = x^2 \).

This holds true for any non-negative real number, making the result accurate and useful for further mathematical operations. Simplifying the variable expressions helps in maintaining a straightforward approach and produces tidy and understandable results, crucial for tackling more advanced algebra problems.