Find the Distance Between Two Points (4 square root of 3, square root of 7) , (- square root of 3,5 square root of 7)

$(4 \sqrt{3}, \sqrt{7})$ , $(- \sqrt{3}, 5 \sqrt{7})$

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.

$\sqrt{{((- \sqrt{3}) - 4 \sqrt{3})}^{2} + {(5 \sqrt{7} - \sqrt{7})}^{2}}$

Step 3

Simplify.

Subtract $4 \sqrt{3}$ from $- \sqrt{3}$ .

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

Simplify the expression.

Apply the product rule to $- 5 \sqrt{3}$ .

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

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

$\sqrt{25 {\sqrt{3}}^{2} + {(5 \sqrt{7} - \sqrt{7})}^{2}}$

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

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

$\sqrt{25 {(3^{\frac{1}{2}})}^{2} + {(5 \sqrt{7} - \sqrt{7})}^{2}}$

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

$\sqrt{25 \cdot 3^{\frac{1}{2} \cdot 2} + {(5 \sqrt{7} - \sqrt{7})}^{2}}$

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

$\sqrt{25 \cdot 3^{\frac{2}{2}} + {(5 \sqrt{7} - \sqrt{7})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{25 \cdot 3^{\frac{2}{2}} + {(5 \sqrt{7} - \sqrt{7})}^{2}}$

Rewrite the expression.

$\sqrt{25 \cdot 3^{1} + {(5 \sqrt{7} - \sqrt{7})}^{2}}$

Evaluate the exponent.

$\sqrt{25 \cdot 3 + {(5 \sqrt{7} - \sqrt{7})}^{2}}$

Multiply $25$ by $3$ .

$\sqrt{75 + {(5 \sqrt{7} - \sqrt{7})}^{2}}$

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

$\sqrt{75 + {(4 \sqrt{7})}^{2}}$

Simplify the expression.

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

$\sqrt{75 + 4^{2} {\sqrt{7}}^{2}}$

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

$\sqrt{75 + 16 {\sqrt{7}}^{2}}$

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

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

$\sqrt{75 + 16 {(7^{\frac{1}{2}})}^{2}}$

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

$\sqrt{75 + 16 \cdot 7^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{75 + 16 \cdot 7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{75 + 16 \cdot 7^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{75 + 16 \cdot 7^{1}}$

Evaluate the exponent.

$\sqrt{75 + 16 \cdot 7}$

Simplify the expression.

Multiply $16$ by $7$ .

$\sqrt{75 + 112}$

Add $75$ and $112$ .

$\sqrt{187}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{187}$

Decimal Form:

$13.67479433 \dots$

Step 5

Find the Distance Between Two Points (4 square root of 3, square root of 7) , (- square root of 3,5 square root of 7)

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