Find the distance $$d$$ between $$Z_1=-{8}+i$$ and $$Z_2=-{1}+{i}$$ (Express your answer in exact terms and simplify, if needed.)



<div id="id_cont_display" class="markdown-body editormd-html-preview"><textarea>Find the distance $$d$$ between $$Z_1=-{8}+i$$ and $$Z_2=-{1}+{i}$$ (Express your answer in exact terms and simplify, if needed.)

</textarea>Find the distance <math><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>d between <math><semantics><mrow><msub><mi>Z</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mrow><mn>8</mn></mrow><mo>+</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">Z_1=-{8}+i</annotation></semantics></math>Z​1​​=−8+i and <math><semantics><mrow><msub><mi>Z</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mrow><mn>1</mn></mrow><mo>+</mo><mrow><mi>i</mi></mrow></mrow><annotation encoding="application/x-tex">Z_2=-{1}+{i}</annotation></semantics></math>Z​2​​=−1+i (Express your answer in exact terms and simplify, if needed.)
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<div id="id_solution_display" class="markdown-body editormd-html-preview"><textarea>Ginen that, $$Z_1=-8+1, Z_2=-1+i$$

The distance between $$Z_1$$ and $$Z_2=d=\sqrt{(-8+1)^{2}+(1-1)^{2}}=\sqrt{7^{2}}=7$$ unit

[By using the formula, $$\sqrt{(x_1-x_2)^{2}+(y_1- y_2)^{2}}$$ where the point are $$x_1+i y_1 \ x_2+i y_2$$]

</textarea>Ginen that, <math><semantics><mrow><msub><mi>Z</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><mn>8</mn><mo>+</mo><mn>1</mn><mo separator="true">,</mo><msub><mi>Z</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">Z_1=-8+1, Z_2=-1+i</annotation></semantics></math>Z​1​​=−8+1,Z​2​​=−1+i
The distance between <math><semantics><mrow><msub><mi>Z</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">Z_1</annotation></semantics></math>Z​1​​ and <math><semantics><mrow><msub><mi>Z</mi><mn>2</mn></msub><mo>=</mo><mi>d</mi><mo>=</mo><msqrt><mrow><mo>(</mo><mo>−</mo><mn>8</mn><mo>+</mo><mn>1</mn><msup><mo>)</mo><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mn>1</mn><msup><mo>)</mo><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo>=</mo><msqrt><mrow><msup><mn>7</mn><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">Z_2=d=\sqrt{(-8+1)^{2}+(1-1)^{2}}=\sqrt{7^{2}}=7</annotation></semantics></math>Z​2​​=d=√​(−8+1)​2​​+(1−1)​2​​​​​=√​7​2​​​​​=7 unit
[By using the formula, <math><semantics><mrow><msqrt><mrow><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msup><mo>)</mo><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>(</mo><msub><mi>y</mi><mn>1</mn></msub><mo>−</mo><msub><mi>y</mi><mn>2</mn></msub><msup><mo>)</mo><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{(x_1-x_2)^{2}+(y_1- y_2)^{2}}</annotation></semantics></math>√​(x​1​​−x​2​​)​2​​+(y​1​​−y​2​​)​2​​​​​ where the point are <math><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>i</mi><msub><mi>y</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mi>i</mi><msub><mi>y</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1+i y_1 \ x_2+i y_2</annotation></semantics></math>x​1​​+iy​1​​&nbsp;x​2​​+iy​2​​]
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