which two numbers does $\sqrt{61}$ lie between on a number line?

7.7 and 7.8
7.8 and 7.9
7.9 and 8.0
8.0 and 8.1

Question

which two numbers does $\sqrt{61}$ lie between on a number line?

7.7 and 7.8
7.8 and 7.9
7.9 and 8.0
8.0 and 8.1

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Square Root Estimation" ], "new_concepts": [], "current_concepts": [ "Square Root Estimation", "Number Line Representation" ] } ## Determine the perfect squares surrounding the radicand $$ 7^2 = 49 < 61 < 64 = 8^2 \implies 7 < \sqrt{61} < 8 $$ ## Calculate the squares of the boundary options $$ \begin{aligned} 7.7^2 &= 59.29 \ 7.8^2 &= 60.84 \ 7.9^2 &= 62.41 \ 8.0^2 &= 64.00 \end{aligned} $$ ## Compare the radicand to the squared boundaries $$ \begin{aligned} 60.84 &< 61 < 62.41 \ 7.8^2 &< 61 < 7.9^2 \ 7.8 &< \sqrt{61} < 7.9 \end{aligned} $$

7.7 and 7.8

7.8 and 7.9

7.9 and 8.0

8.0 and 8.1

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Square Root Estimation" ] }

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QUESTION IMAGE

Question

Explanation:

Determine the perfect squares surrounding the radicand

Calculate the squares of the boundary options

Compare the radicand to the squared boundaries

Answer: