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Grade 11 · unit review

Review: Algebra2: Radicals and Complex Values

Root expressions can be simplified through perfect-power factors and rational exponents. Squaring equations can introduce extraneous candidates, while complex arithmetic extends the number system using an imaginary unit with square negative one.

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Bring the unit skills together

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. Simplify √588 exactly.
Need a hint?
  • A nonnegative radical can be simplified by separating a perfect-square factor.
  • Keep the factor outside positive for these principal square roots.
2. Evaluate 343^(2/3) as a real number.
Need a hint?
  • For the positive bases here, an exponent m/n means take an nth root and then an mth power.
  • The denominator names the root and the numerator names the power.
3. Solve √(x+7)=−8 over the reals.
Need a hint?
  • A principal square root cannot equal a negative real number.
  • Squaring can enlarge a solution set, so every candidate needs substitution back into the original.
4. Evaluate (7+i)(7−i), using i²=−1.
Need a hint?
  • A conjugate pair differs only in its imaginary-part sign.
  • Treat i as a symbol with the specific rule i²=−1.

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. Simplify √972 exactly. Candidate answers: (A) 108√3 (B) 18√3 (C) 18√6 (D) 9√3. 2. Evaluate 729^(2/3) as a real number. Candidate answers: (A) 162 (B) 80 (C) 81 (D) 82. 3. Solve √(x+9)=−10 over the reals. Candidate answers: (A) No real solution (B) x=19 (C) x=91 (D) x=−109. 4. Evaluate (9+i)(9−i), using i²=−1. Candidate answers: (A) 164 (B) 81 (C) 82 (D) 83.

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. 18√3. 972=(18)²·3. Taking the positive root of the perfect-square factor gives 18√3, whose remaining radicand has no square factor. 2. 81. Take the cube root first: (9³)^(1/3)=9. Then square: 9²=81. The denominator of the exponent tells which root to take. 3. No real solution. The principal square root is nonnegative, whereas −10<0. Squaring produces x=91, but it does not satisfy the original sign condition. Checking rejects this extraneous candidate. 4. 82. Distribute: 9²−9i+9i−i². The imaginary terms cancel, and −i²=1, so the result is 81+1=82.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.