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Match the square root or cube root to their simplified version.(4 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. 40−−√ 125−−−√ 625−−−√3 40−−√3 210−−√55–√325–√355–√

1 answer

Let's simplify each expression step-by-step.

  1. Square Root of 40: \[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \]

  2. Square Root of 125: \[ \sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5} \]

  3. Cube Root of 625: \[ \sqrt[3]{625} = \sqrt[3]{25^2} = 25^{\frac{2}{3}} = 5^2 \cdot 5^{\frac{2}{3}} = 5^{\frac{6}{3}} \cdot 5^{\frac{2}{3}} = 5^{\frac{8}{3}} = 5^{2} \sqrt[3]{5} = 25 \sqrt[3]{5} \]

  4. Square Root of 40 multiplied by cube root of 3: \[ \sqrt{40} \cdot \sqrt[3]{3} = 2\sqrt{10} \cdot \sqrt[3]{3}. \]

  5. Square Root of 210: \[ \sqrt{210} = \sqrt{2 \times 3 \times 5 \times 7} \] This does not simplify further.

  6. Square Root of 55: \[ \sqrt{55} = \sqrt{5 \times 11} \] This does not simplify further.

  7. Square Root of 325: \[ \sqrt{325} = \sqrt{25 \times 13} = 5\sqrt{13} \]

  8. Square Root of 355: \[ \sqrt{355} = \sqrt{5 \times 71} \] This does not simplify further.

The simplified versions thus are:

  • \(\sqrt{40} \) = \(2\sqrt{10}\)
  • \(\sqrt{125}\) = \(5\sqrt{5}\)
  • \(\sqrt[3]{625}\) = \(25\sqrt[3]{5}\)
  • \(\sqrt{40} \cdot \sqrt[3]{3}\) = \(2\sqrt{10}\cdot \sqrt[3]{3}\)

With this information, you can now match the expressions to their simplified forms as necessary!