Precalculus Examples

$(\sqrt{125}, \sqrt{20})$ , $(7, 2 \sqrt{5})$

Step 1

Rewrite $125$ as $5^{2} \cdot 5$ .

Tap for more steps...

Step 1.1

Factor $25$ out of $125$ .

$(\sqrt{25 (5)}, \sqrt{20}) - (7, 2 \sqrt{5})$

Step 1.2

Rewrite $25$ as $5^{2}$ .

$(\sqrt{5^{2} \cdot 5}, \sqrt{20}) - (7, 2 \sqrt{5})$

Step 2

Pull terms out from under the radical.

$(5 \sqrt{5}, \sqrt{20}) - (7, 2 \sqrt{5})$

Step 3

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 3.1

Factor $4$ out of $20$ .

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

Step 3.2

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

Rewrite ${(7 - 5 \sqrt{5})}^{2}$ as $(7 - 5 \sqrt{5}) (7 - 5 \sqrt{5})$ .

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

Step 7.2

Expand $(7 - 5 \sqrt{5}) (7 - 5 \sqrt{5})$ using the FOIL Method.

Tap for more steps...

Step 7.2.1

Apply the distributive property.

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

Step 7.2.2

Apply the distributive property.

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

Step 7.2.3

Apply the distributive property.

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

Step 7.3

Simplify and combine like terms.

Tap for more steps...

Step 7.3.1

Simplify each term.

Tap for more steps...

Step 7.3.1.1

Multiply $7$ by $7$ .

$\sqrt{49 + 7 (- 5 \sqrt{5}) - 5 \sqrt{5} \cdot 7 - 5 \sqrt{5} (- 5 \sqrt{5}) + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.2

Multiply $- 5$ by $7$ .

$\sqrt{49 - 35 \sqrt{5} - 5 \sqrt{5} \cdot 7 - 5 \sqrt{5} (- 5 \sqrt{5}) + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.3

Multiply $7$ by $- 5$ .

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} - 5 \sqrt{5} (- 5 \sqrt{5}) + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.4

Multiply $- 5 \sqrt{5} (- 5 \sqrt{5})$ .

Tap for more steps...

Step 7.3.1.4.1

Multiply $- 5$ by $- 5$ .

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 \sqrt{5} \sqrt{5} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 ({\sqrt{5}}^{1} \sqrt{5}) + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 {\sqrt{5}}^{1 + 1} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.4.5

Add $1$ and $1$ .

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 {\sqrt{5}}^{2} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.5

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

Tap for more steps...

Step 7.3.1.5.1

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

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 {(5^{\frac{1}{2}})}^{2} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.5.2

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

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 \cdot 5^{\frac{1}{2} \cdot 2} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.5.3

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

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 \cdot 5^{\frac{2}{2}} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.3.1.5.4.1

Cancel the common factor.

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 \cdot 5^{\frac{2}{2}} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.5.4.2

Rewrite the expression.

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 \cdot 5^{1} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.5.5

Evaluate the exponent.

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 25 \cdot 5 + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.1.6

Multiply $25$ by $5$ .

$\sqrt{49 - 35 \sqrt{5} - 35 \sqrt{5} + 125 + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.2

Add $49$ and $125$ .

$\sqrt{174 - 35 \sqrt{5} - 35 \sqrt{5} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.3.3

Subtract $35 \sqrt{5}$ from $- 35 \sqrt{5}$ .

$\sqrt{174 - 70 \sqrt{5} + {(2 \sqrt{5} - 2 \sqrt{5})}^{2}}$

Step 7.4

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

$\sqrt{174 - 70 \sqrt{5} + 0^{2}}$

Step 7.5

Simplify the expression.

Tap for more steps...

Step 7.5.1

Raising $0$ to any positive power yields $0$ .

$\sqrt{174 - 70 \sqrt{5} + 0}$

Step 7.5.2

Add $174 - 70 \sqrt{5}$ and $0$ .

$\sqrt{174 - 70 \sqrt{5}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{174 - 70 \sqrt{5}}$

Decimal Form:

$4.18033988 \dots$

Step 9