Find the Distance Between Two Points (2 square root of 2, square root of 7) , (- square root of 2,3 square root of 7)

$(2 \sqrt{2}, \sqrt{7})$ , $(- \sqrt{2}, 3 \sqrt{7})$

Step 1

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Step 2

Substitute the actual values of the points into the distance formula.

$\sqrt{{((- \sqrt{2}) - 2 \sqrt{2})}^{2} + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Step 3

Simplify.

Subtract $2 \sqrt{2}$ from $- \sqrt{2}$ .

$\sqrt{{(- 3 \sqrt{2})}^{2} + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Simplify the expression.

Apply the product rule to $- 3 \sqrt{2}$ .

$\sqrt{{(- 3)}^{2} {\sqrt{2}}^{2} + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Raise $- 3$ to the power of $2$ .

$\sqrt{9 {\sqrt{2}}^{2} + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sqrt{9 {(2^{\frac{1}{2}})}^{2} + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{9 \cdot 2^{\frac{1}{2} \cdot 2} + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{9 \cdot 2^{\frac{2}{2}} + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{9 \cdot 2^{\frac{2}{2}} + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Rewrite the expression.

$\sqrt{9 \cdot 2^{1} + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Evaluate the exponent.

$\sqrt{9 \cdot 2 + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Multiply $9$ by $2$ .

$\sqrt{18 + {(3 \sqrt{7} - \sqrt{7})}^{2}}$

Subtract $\sqrt{7}$ from $3 \sqrt{7}$ .

$\sqrt{18 + {(2 \sqrt{7})}^{2}}$

Simplify the expression.

Apply the product rule to $2 \sqrt{7}$ .

$\sqrt{18 + 2^{2} {\sqrt{7}}^{2}}$

Raise $2$ to the power of $2$ .

$\sqrt{18 + 4 {\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{18 + 4 {(7^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{18 + 4 \cdot 7^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{18 + 4 \cdot 7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{18 + 4 \cdot 7^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{18 + 4 \cdot 7^{1}}$

Evaluate the exponent.

$\sqrt{18 + 4 \cdot 7}$

Simplify the expression.

Multiply $4$ by $7$ .

$\sqrt{18 + 28}$

Add $18$ and $28$ .

$\sqrt{46}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{46}$

Decimal Form:

$6.78232998 \dots$

Step 5

Find the Distance Between Two Points (2 square root of 2, square root of 7) , (- square root of 2,3 square root of 7)

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