Find the Distance Between Two Points (4 square root of 2, square root of 5) and (- square root of 2,5 square root of 5)

$(4 \sqrt{2}, \sqrt{5})$ and $(- \sqrt{2}, 5 \sqrt{5})$

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

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

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

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

Simplify.

Subtract $4 \sqrt{2}$ from $- \sqrt{2}$ .

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

Simplify the expression.

Apply the product rule to $- 5 \sqrt{2}$ .

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

Raise $- 5$ to the power of $2$ .

$\sqrt{25 {\sqrt{2}}^{2} + {(5 \sqrt{5} - \sqrt{5})}^{2}}$

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

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

$\sqrt{25 {(2^{\frac{1}{2}})}^{2} + {(5 \sqrt{5} - \sqrt{5})}^{2}}$

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

$\sqrt{25 \cdot 2^{\frac{1}{2} \cdot 2} + {(5 \sqrt{5} - \sqrt{5})}^{2}}$

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

$\sqrt{25 \cdot 2^{\frac{2}{2}} + {(5 \sqrt{5} - \sqrt{5})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{25 \cdot 2^{\frac{2}{2}} + {(5 \sqrt{5} - \sqrt{5})}^{2}}$

Rewrite the expression.

$\sqrt{25 \cdot 2^{1} + {(5 \sqrt{5} - \sqrt{5})}^{2}}$

Evaluate the exponent.

$\sqrt{25 \cdot 2 + {(5 \sqrt{5} - \sqrt{5})}^{2}}$

Multiply $25$ by $2$ .

$\sqrt{50 + {(5 \sqrt{5} - \sqrt{5})}^{2}}$

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

$\sqrt{50 + {(4 \sqrt{5})}^{2}}$

Simplify the expression.

Apply the product rule to $4 \sqrt{5}$ .

$\sqrt{50 + 4^{2} {\sqrt{5}}^{2}}$

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

$\sqrt{50 + 16 {\sqrt{5}}^{2}}$

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

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

$\sqrt{50 + 16 {(5^{\frac{1}{2}})}^{2}}$

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

$\sqrt{50 + 16 \cdot 5^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{50 + 16 \cdot 5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{50 + 16 \cdot 5^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{50 + 16 \cdot 5^{1}}$

Evaluate the exponent.

$\sqrt{50 + 16 \cdot 5}$

Simplify the expression.

Multiply $16$ by $5$ .

$\sqrt{50 + 80}$

Add $50$ and $80$ .

$\sqrt{130}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt{130}$

Decimal Form:

$11.40175425 \dots$

Find the Distance Between Two Points (4 square root of 2, square root of 5) and (- square root of 2,5 square root of 5)

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