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Grade 8 · unit review

Review: Real Roots and Approximation

Roots reverse powers, but their exact values and decimal approximations must be distinguished. Integer squares provide reliable bounds, and odd roots retain negative signs while principal square roots are nonnegative.

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Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Between which consecutive whole numbers is √50?
Need a hint?
  • To locate a positive square root, compare its radicand with nearby perfect squares.
  • Compare squares of candidate endpoints, not endpoints directly with the radicand.
2. Which supplied value, among fractions, repeating decimals and roots of 49 or 50, is irrational?
Need a hint?
  • Fractions and terminating or repeating decimals are rational.
  • A calculator display never proves irrationality by itself.
3. Approximate √51 to the nearest tenth; a calculator may be used.
Need a hint?
  • A decimal root approximation can be checked against whole-number bounds before rounding.
  • Round only after calculating enough digits, and retain a calculator value through intermediate work when needed.
4. Find the real cube root of −343.
Need a hint?
  • A real cube root is the unique real number whose cube is the radicand.
  • Do not apply the nonnegative principal-square-root convention to cube roots.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Between which consecutive whole numbers is √82? Candidate answers: (A) 10 and 11 (B) 8 and 9 (C) 81 and 82 (D) 9 and 10. 2. Which supplied value, among fractions, repeating decimals and roots of 81 or 82, is irrational? Candidate answers: (A) 0.999… with the block 9 repeating (B) 9/4 (C) √81 (D) √82. 3. Approximate √83 to the nearest tenth; a calculator may be used. Candidate answers: (A) 10 (B) 41.5 (C) 9 (D) 9.1. 4. Find the real cube root of −729. Candidate answers: (A) -729 (B) -81 (C) -9 (D) 9.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 9 and 10. 9²=81<82, while 10²=100>82. The positive square root therefore lies strictly between 9 and 10. 2. √82. 82 lies between consecutive integer squares, so √82 is not a rational number. A fraction, a repeating decimal and the square root of a perfect square are rational. 3. 9.1. The exact root lies between 9 and 10. Its decimal value is approximately 9.110434; rounding the hundredths place gives 9.1. 4. -9. Because (−9)³=−729, the cube root is −9. Odd powers preserve the sign, unlike even powers.

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