Problem 62 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt[3]{\sqrt{64 x^{6} y^{12}}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[3]{\sqrt{64 x^{6} y^{12}}} \) is \(2x y^{2}\).

Step 1: Identify the Innermost Radical

In the given equation, you can see the square root inside the cube root. This is the first part of the expression that needs to be simplified.\[\sqrt[3]{\sqrt{64 x^{6} y^{12}}}\]

Step 2: Simplify the Inner Square Root

Start by simplifying the inner square root. The square root of \(64 x^{6} y^{12}\) is calculated by taking the square root of 64 and the square root of each variable part. The square root of 64 is 8, and for the variables, the square root of \(x^{6}\) is \(x^{3}\) and the square root of \(y^{12}\) is \(y^{6}\). This simplification results in:\[\sqrt[3]{8 x^{3} y^{6}}\]

Step 3: Simplify the Outer Cube Root

Next, simplify the outer cube root. The cube root of \(8 x^{3} y^{6}\) is calculated by taking the cube root of 8 and the cube root of each variable part. The cube root of 8 is 2, and for the variables, the cube root of \(x^{3}\) is \(x\) and the cube root of \(y^{6}\) is \(y^{2}\). This simplification results in:\[2x y^{2}\]

Key Concepts

Cube Roots

Cube roots are all about finding a number which, when multiplied by itself three times, gives the original number. For instance, the cube root of 8 is 2, because if you multiply 2 by itself three times (2 \( \times \) 2 \( \times \) 2), you get 8. In mathematical notation, the cube root is expressed using a radical with an index of 3, represented as \( \sqrt[3]{a} \), where \( a \) is the number you're finding the cube root of.

• To calculate the cube root of a variable raised to a power like \( x^n \), divide the exponent \( n \) by 3. For example, \( \sqrt[3]{x^6} = x^{6/3} = x^2 \).
• This concept is useful in simplifying expressions, especially when working with nested radicals, such as in our exercise.

Cube roots form the foundation for working with three-dimensional volumes and other geometric concepts, as they reflect a natural cube's properties.

Square Roots

Square roots are one of the most fundamental concepts in mathematics, representing a value that, when multiplied by itself, equals the original number. For instance, the square root of 64 is 8, because 8 multiplied by 8 equals 64. The square root is expressed using a radical sign, \( \sqrt{a} \), where \( a \) is the desired number.

• The square root is especially straightforward when dealing with perfect squares, like 64, 100, or 144.
• When dealing with variables, finding the square root involves halving the exponent of the variable: \( \sqrt{x^n} = x^{n/2} \).

Remembering these rules helps you break down complex expressions into simpler components, making it easier to manage equations and solve problems.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without altering their value. This process often includes combining like terms, reducing fractions, and removing unnecessary radicals or exponents.

• The aim is to make an expression as manageable as possible, whether you're dealing with numerical values or algebraic terms.
• In contexts where radicals are involved, this might involve taking square or cube roots to remove them from under the radical sign.
• If fractions or rational expressions are present, rationalization might be necessary to eliminate radical expressions from the denominator.

By simplifying expressions, you not only make the arithmetic easier but also clarify the relationship between different components of the equation, thus aiding in deeper understanding and solving more complex problems.