Find the Distance Between Two Points (5 square root of 7,-3 square root of 11) , (9 square root of 7,-2 square root of 11)

$(5 \sqrt{7}, - 3 \sqrt{11})$ , $(9 \sqrt{7}, - 2 \sqrt{11})$

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.

$\sqrt{{(9 \sqrt{7} - 5 \sqrt{7})}^{2} + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

Step 3

Simplify.

Subtract $5 \sqrt{7}$ from $9 \sqrt{7}$ .

$\sqrt{{(4 \sqrt{7})}^{2} + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

Simplify the expression.

Apply the product rule to $4 \sqrt{7}$ .

$\sqrt{4^{2} {\sqrt{7}}^{2} + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

Raise $4$ to the power of $2$ .

$\sqrt{16 {\sqrt{7}}^{2} + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

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

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

$\sqrt{16 {(7^{\frac{1}{2}})}^{2} + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

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

$\sqrt{16 \cdot 7^{\frac{1}{2} \cdot 2} + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

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

$\sqrt{16 \cdot 7^{\frac{2}{2}} + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{16 \cdot 7^{\frac{2}{2}} + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

Rewrite the expression.

$\sqrt{16 \cdot 7^{1} + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

Evaluate the exponent.

$\sqrt{16 \cdot 7 + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

Multiply.

Multiply $16$ by $7$ .

$\sqrt{112 + {((- 2 \sqrt{11}) - (- 3 \sqrt{11}))}^{2}}$

Multiply $- 3$ by $- 1$ .

$\sqrt{112 + {(- 2 \sqrt{11} + 3 \sqrt{11})}^{2}}$

Add $- 2 \sqrt{11}$ and $3 \sqrt{11}$ .

$\sqrt{112 + {\sqrt{11}}^{2}}$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\sqrt{112 + {(11^{\frac{1}{2}})}^{2}}$

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

$\sqrt{112 + 11^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{112 + 11^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{112 + 11^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{112 + 11^{1}}$

Evaluate the exponent.

$\sqrt{112 + 11}$

Add $112$ and $11$ .

$\sqrt{123}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{123}$

Decimal Form:

$11.09053650 \dots$

Step 5

Find the Distance Between Two Points (5 square root of 7,-3 square root of 11) , (9 square root of 7,-2 square root of 11)

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