Math Problem Statement

(2 radicl din 2 -3a) (2radical din 2 +3a) ÷(x/3+2y)³

Solution

The given expression is:

(223a)(22+3a)(x3+2y)3\frac{(2\sqrt{2} - 3a)(2\sqrt{2} + 3a)}{\left(\frac{x}{3} + 2y\right)^3}

Let's simplify this step by step.

Step 1: Simplifying the numerator

The numerator (223a)(22+3a)(2\sqrt{2} - 3a)(2\sqrt{2} + 3a) is a difference of squares, which follows the identity:

(ab)(a+b)=a2b2(a - b)(a + b) = a^2 - b^2

Here, a=22a = 2\sqrt{2} and b=3ab = 3a. Applying the identity:

(22)2(3a)2(2\sqrt{2})^2 - (3a)^2

Now calculate each square:

(22)2=22×(2)2=4×2=8(2\sqrt{2})^2 = 2^2 \times (\sqrt{2})^2 = 4 \times 2 = 8 (3a)2=9a2(3a)^2 = 9a^2

Thus, the numerator becomes:

89a28 - 9a^2

Step 2: Writing the full expression

Now the expression becomes:

89a2(x3+2y)3\frac{8 - 9a^2}{\left(\frac{x}{3} + 2y\right)^3}

Final result:

89a2(x3+2y)3\frac{8 - 9a^2}{\left(\frac{x}{3} + 2y\right)^3}

Would you like to explore further details or ask any questions about this expression?

Here are 5 related questions:

  1. How can you simplify a binomial raised to a power, such as (x+y)3(x + y)^3?
  2. What is the difference of squares identity and how is it applied?
  3. How do you simplify square roots in expressions like 222\sqrt{2}?
  4. What happens if you expand the denominator (x3+2y)3\left(\frac{x}{3} + 2y\right)^3?
  5. How would this expression change if additional variables are introduced in the numerator?

Tip: When working with square roots and powers, always check if you can apply identities like difference of squares for faster simplification!

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Math Problem Analysis

Mathematical Concepts

Algebra
Radicals
Difference of Squares
Exponentiation

Formulas

(a - b)(a + b) = a^2 - b^2 (Difference of Squares)
(a^b) = (a × a × ... × a) b times (Exponentiation)

Theorems

Difference of Squares

Suitable Grade Level

Grades 9-12