Question

Simplify the expressions, assuming all variables are positive. $$ \frac{\sqrt[3]{96 x^{7} y^{8}}}{\sqrt[3]{3 x^{4} y}} $$

Short answer

Question: Simplify the expression $$\frac{\sqrt[3]{96 x^{7} y^{8}}}{\sqrt[3]{3 x^{4} y}}$$, assuming all variables are positive. Answer: The simplified expression is $$2\sqrt[3]{2}x\sqrt[3]{x^{2}}y^2\sqrt[3]{y}$$.

Step-by-step solution

Write the given expression

$$ \frac{\sqrt[3]{96 x^{7} y^{8}}}{\sqrt[3]{3 x^{4} y}} $$ Step 2: Factorize the terms inside the cube roots

Factorize the numbers in the expression

$$ \frac{\sqrt[3]{(2^5 \cdot 3) x^{7} y^{8}}}{\sqrt[3]{3 x^{4} y}} $$ Step 3: Use properties of exponents and radicals

Simplify the expression by using quotient of powers property and the property of radicals

$$ = \sqrt[3]{\frac{2^5 \cdot 3^1 \cdot x^7 \cdot y^8}{3^1 \cdot x^4 \cdot y^1}} $$ Step 4: Simplify each fraction

Cancel out the common terms and simplify each fraction

$$ = \sqrt[3]{2^5 \cdot x^3 \cdot y^7} $$ Step 5: Use the power of a product property

Use the property of radicals to simplify the expression

$$ = 2\sqrt[3]{2} \cdot x\sqrt[3]{x^{2}} \cdot y^2\sqrt[3]{y} $$ Step 6: Present the final expression

Write the final simplified expression

$$ 2\sqrt[3]{2}x\sqrt[3]{x^{2}}y^2\sqrt[3]{y} $$

Key concepts

Properties of Exponents

The properties of exponents are essential rules that help make complex expressions easier to manage. Here are some key properties related to our exercise:

• Quotient of Powers Property: When dividing terms with the same base, subtract the exponents. For instance, \( \frac{x^7}{x^4} = x^{7-4} = x^3 \).
• Power of a Product Property: When a term with multiple factors is raised to an exponent, the exponent can be distributed to each factor. For example, \((xy)^8 = x^8y^8\).
• Power of a Power Property: When raising an exponent to another exponent, multiply them. For example, \((x^3)^2 = x^{3 \times 2} = x^6\).

These properties help in simplifying expressions and solving this type of algebraic problem more efficiently. They transform complicated exponents into simpler forms that are easier to handle.

Cube Roots

Cube roots are one of the important types of radicals used in algebra. The cube root of a number \( a \), denoted as \( \sqrt[3]{a} \), is a value that, when cubed, gives \( a \).

When dealing with cube roots:

• Understanding Factorization: Breaking down numbers into their prime factors can make finding cube roots simpler. For example, figuring out \( \sqrt[3]{96} \) involves factorizing 96 into \( 2^5 \times 3 \).
• Applying to Variables: Cube roots apply not only to numerical coefficients but also to variables. If you have \( x^7 \), its cube root would be expressed as \( \sqrt[3]{x^7} = x^{7/3} \).

The cube root helps simplify expressions by reducing the complexity of the roots when variables and coefficients are present.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Simplifying them involves applying various mathematical rules and operations. In the given problem, we are working with an expression under cube roots that require simplification:

• Decomposing Expressions: Breaking apart the expression \( \frac{\sqrt[3]{96 x^7 y^8}}{\sqrt[3]{3 x^4 y}} \) involves separating it into its base components.
• Simplification Process: First, factorize each part inside the cube roots. For example, breaking down \( 96 x^7 y^8 \) into \( 2^5 \times 3 \times x^7 \times y^8 \) allows you to apply the cube root effectively.
• Reducing Complexity: Use properties, such as canceling out common terms \( 3^1 \), \( x^4 \), and \( y^1 \) in the expression, to arrive at a simpler form \( 2\sqrt[3]{2}x\sqrt[3]{x^2}y^2\sqrt[3]{y} \).

Simplifying algebraic expressions can make complex problems more digestible—especially when dealing with radicals and exponents.