Math Problem Statement

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

Solution

You provided the following expression:

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

Let's break this down step-by-step.

Step 1: Simplify the numerator

The numerator is a difference of squares, which follows the formula:

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

In your case:

  • a=22a = 2\sqrt{2}
  • b=3ab = 3a

Applying the formula:

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

Simplify each term:

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

So, the simplified numerator is:

89a28 - 9a^2

Step 2: Denominator stays the same

The denominator is:

(x3+2y)3\left(\frac{x}{3} + 2y\right)^3

Step 3: Final expression

After simplifying, the expression becomes:

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

Would you like further details on any part of this solution?

5 Relative Questions:

  1. Can you verify what range of values for aa or x,yx, y will simplify the expression?
  2. How does this expression behave when x=0x = 0 or y=0y = 0?
  3. Would this be easier to work with if the denominator was expanded or simplified further?
  4. What are the domain restrictions for x,y,ax, y, a to keep this expression defined?
  5. How would this expression change if powers or roots were involved in the denominator?

Tip:

Always remember to check for domain restrictions when variables appear in the denominator!

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Math Problem Analysis

Mathematical Concepts

Algebra
Difference of Squares
Exponentiation

Formulas

Difference of Squares: (a - b)(a + b) = a^2 - b^2
Exponentiation: (a + b)^n

Theorems

Difference of Squares Theorem

Suitable Grade Level

Grades 9-12