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Understanding cube roots is key to simplifying algebraic expressions, especially when dealing with radical expressions. A cube root asks the question: what number, when multiplied by itself twice, gives the original number? Mathematically, if you have a number \( a \), its cube root is written as \( \sqrt[3]{a} \).

• For example, \( \sqrt[3]{8} = 2 \) because \( 2 \times 2 \times 2 = 8 \).
• Similarly, \( \sqrt[3]{27} = 3 \) because \( 3 \times 3 \times 3 = 27 \).

In the algebraic expression \( \sqrt[3]{\frac{81 x^{8}}{8 y^{15}}} \), focus on both the numerator and denominator. Break them into two separate parts: \( \sqrt[3]{81x^{8}} \) and \( \sqrt[3]{8y^{15}} \).
To simplify cube roots of variables and numbers like \( 81 \) and \( 8 \), decompose them into their prime factors and variables into their base parts. This breaks complex expressions into manageable terms, making it easier to simplify.

Simplifying algebraic expressions efficiently requires breaking down the expression into manageable parts. Begin by assessing the expression and identifying any radical expressions or exponential terms that can be simplified.When you encounter an expression such as \( \sqrt[3]{81x^{8}} \), you must:

• First, simplify the numerical part. Calculate \( \sqrt[3]{81} \), which simplifies to approximately \( 4.326 \).
• Next, simplify the variable with an exponent by converting it to a fractional exponent, resulting in \( x^{8/3} \).

Apply this process to the denominator as well, \( \sqrt[3]{8y^{15}} \):

• Cube root of \( 8 \) is \( 2 \).
• Cube root of \( y^{15} \) is \( y^{5} \), because \( 15 \div 3 = 5 \).

Understanding how to simplify each component allows for correctly simplifying complex expressions, ensuring accurate solutions.

Exponents are fundamental in algebra and involve numbers raised to the power indicating how many times to multiply the base by itself. An expression like \( x^n \) implies that \( x \) is multiplied by itself \( n \) times. When combined with roots, exponents become fractional.Fractional exponents are pivotal in simplifying expressions involving roots:

• For example, \( x^{8/3} \) indicates the cube root of \( x^8 \). Here, \( 8/3 \) suggests that \( x \) is raised to the 8th power first, and then you take this value’s cube root.
• The expression \( y^{5} \), derived from \( y^{15/3} \), illustrates reducing the power by dividing it by the root (\( 3 \) in this case).

These principles apply when simplifying the overall expression \( \frac{2.163x^{8/3}}{y^{5}} \), achieved by leveraging properties of exponents and roots. Proper manipulation of these exponents allows you to simplify even seemingly complicated expressions with ease.