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

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

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.

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

Step 3

Simplify.

Subtract $3 \sqrt{2}$ from $\sqrt{2}$ .

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

Simplify the expression.

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

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

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

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

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

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

$\sqrt{4 {(2^{\frac{1}{2}})}^{2} + {((- \sqrt{3}) - 7 \sqrt{3})}^{2}}$

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

$\sqrt{4 \cdot 2^{\frac{1}{2} \cdot 2} + {((- \sqrt{3}) - 7 \sqrt{3})}^{2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{4 \cdot 2^{1} + {((- \sqrt{3}) - 7 \sqrt{3})}^{2}}$

Evaluate the exponent.

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

Multiply $4$ by $2$ .

$\sqrt{8 + {((- \sqrt{3}) - 7 \sqrt{3})}^{2}}$

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

$\sqrt{8 + {(- 8 \sqrt{3})}^{2}}$

Simplify the expression.

Apply the product rule to $- 8 \sqrt{3}$ .

$\sqrt{8 + {(- 8)}^{2} {\sqrt{3}}^{2}}$

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

$\sqrt{8 + 64 {\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{8 + 64 {(3^{\frac{1}{2}})}^{2}}$

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

$\sqrt{8 + 64 \cdot 3^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{8 + 64 \cdot 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{8 + 64 \cdot 3^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{8 + 64 \cdot 3^{1}}$

Evaluate the exponent.

$\sqrt{8 + 64 \cdot 3}$

Simplify the expression.

Multiply $64$ by $3$ .

$\sqrt{8 + 192}$

Add $8$ and $192$ .

$\sqrt{200}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$\sqrt{100 (2)}$

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2} \cdot 2}$

Pull terms out from under the radical.

$10 \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt{2}$

Decimal Form:

$14.14213562 \dots$

Step 5

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

Related Questions

Support LCMGCF.com by disabling your adblocker.