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

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

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.

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

Step 3

Simplify.

Subtract $5 \sqrt{2}$ from $\sqrt{2}$ .

$\sqrt{{(- 4 \sqrt{2})}^{2} + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

Simplify the expression.

Apply the product rule to $- 4 \sqrt{2}$ .

$\sqrt{{(- 4)}^{2} {\sqrt{2}}^{2} + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

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

$\sqrt{16 {\sqrt{2}}^{2} + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

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

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

$\sqrt{16 {(2^{\frac{1}{2}})}^{2} + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

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

$\sqrt{16 \cdot 2^{\frac{1}{2} \cdot 2} + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

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

$\sqrt{16 \cdot 2^{\frac{2}{2}} + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{16 \cdot 2^{\frac{2}{2}} + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

Rewrite the expression.

$\sqrt{16 \cdot 2^{1} + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

Evaluate the exponent.

$\sqrt{16 \cdot 2 + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

Multiply $16$ by $2$ .

$\sqrt{32 + {((- \sqrt{5}) - 7 \sqrt{5})}^{2}}$

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

$\sqrt{32 + {(- 8 \sqrt{5})}^{2}}$

Simplify the expression.

Apply the product rule to $- 8 \sqrt{5}$ .

$\sqrt{32 + {(- 8)}^{2} {\sqrt{5}}^{2}}$

Raise $- 8$ to the power of $2$ .

$\sqrt{32 + 64 {\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{32 + 64 {(5^{\frac{1}{2}})}^{2}}$

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

$\sqrt{32 + 64 \cdot 5^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{32 + 64 \cdot 5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{32 + 64 \cdot 5^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{32 + 64 \cdot 5^{1}}$

Evaluate the exponent.

$\sqrt{32 + 64 \cdot 5}$

Simplify the expression.

Multiply $64$ by $5$ .

$\sqrt{32 + 320}$

Add $32$ and $320$ .

$\sqrt{352}$

Rewrite $352$ as $4^{2} \cdot 22$ .

Factor $16$ out of $352$ .

$\sqrt{16 (22)}$

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \cdot 22}$

Pull terms out from under the radical.

$4 \sqrt{22}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{22}$

Decimal Form:

$18.76166303 \dots$

Step 5

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

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