Divide ((2pi)/(4pi^2-1))÷((6pi^3)/(6pi+3))

$\frac{2 π}{4 π^{2} - 1} \div \frac{6 π^{3}}{6 π + 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 π}{4 π^{2} - 1} \cdot \frac{6 π + 3}{6 π^{3}}$

Step 2

Cancel the common factor of $2 π$ .

Factor $2 π$ out of $6 π^{3}$ .

$\frac{2 π}{4 π^{2} - 1} \cdot \frac{6 π + 3}{2 π (3 π^{2})}$

Cancel the common factor.

$\frac{2 π}{4 π^{2} - 1} \cdot \frac{6 π + 3}{2 π (3 π^{2})}$

Rewrite the expression.

$\frac{1}{4 π^{2} - 1} \cdot \frac{6 π + 3}{3 π^{2}}$

Step 3

Multiply $\frac{1}{4 π^{2} - 1}$ by $\frac{6 π + 3}{3 π^{2}}$ .

$\frac{6 π + 3}{(4 π^{2} - 1) (3 π^{2})}$

Step 4

Cancel the common factor of $6 π + 3$ and $3$ .

Factor $3$ out of $6 π$ .

$\frac{3 (2 π) + 3}{(4 π^{2} - 1) (3 π^{2})}$

Factor $3$ out of $3$ .

$\frac{3 (2 π) + 3 \cdot 1}{(4 π^{2} - 1) (3 π^{2})}$

Factor $3$ out of $3 (2 π) + 3 (1)$ .

$\frac{3 (2 π + 1)}{(4 π^{2} - 1) (3 π^{2})}$

Cancel the common factors.

Factor $3$ out of $(4 π^{2} - 1) (3 π^{2})$ .

$\frac{3 (2 π + 1)}{3 ((4 π^{2} - 1) (π^{2}))}$

Cancel the common factor.

$\frac{3 (2 π + 1)}{3 ((4 π^{2} - 1) (π^{2}))}$

Rewrite the expression.

$\frac{2 π + 1}{(4 π^{2} - 1) (π^{2})}$

Step 5

Simplify the denominator.

Rewrite $4 π^{2}$ as ${(2 π)}^{2}$ .

$\frac{2 π + 1}{({(2 π)}^{2} - 1) π^{2}}$

Rewrite $1$ as $1^{2}$ .

$\frac{2 π + 1}{({(2 π)}^{2} - 1^{2}) π^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 π$ and $b = 1$ .

$\frac{2 π + 1}{(2 π + 1) (2 π - 1) π^{2}}$

Step 6

Cancel the common factor of $2 π + 1$ .

Cancel the common factor.

$\frac{2 π + 1}{(2 π + 1) (2 π - 1) π^{2}}$

Rewrite the expression.

$\frac{1}{(2 π - 1) π^{2}}$

Step 7

Reorder factors in $\frac{1}{(2 π - 1) π^{2}}$ .

$\frac{1}{π^{2} (2 π - 1)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{π^{2} (2 π - 1)}$

Decimal Form:

$0.01917804 \dots$

Divide ((2pi)/(4pi^2-1))÷((6pi^3)/(6pi+3))

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