Practice 1 Solutions

Evaluate.

Note

Use the Formula Sheet to help with the rules for the imaginary unit i.

$i^{29}$

$29 \div 4 = 7 R 1 {(i^{4})}^{7} \cdot i 1 \cdot i$

$i^{75}$

$75 \div 4 = 18 R 3 {(i^{4})}^{18} \cdot i^{3} 1 \cdot - i$

– i

$i^{16}$

$16 \div 4 = 4 {(i^{4})}^{4}$

$i^{50}$

$50 \div 4 = 12 R 2 {(i^{4})}^{12} \cdot i^{2} 1 \cdot - 1$

–1

$i^{120}$

$120 \div 4 = 30 {(i^{4})}^{30}$

$i^{101}$

$101 \div 4 = 25 R 1 {(i^{4})}^{25} \cdot i 1 \cdot i$

$i^{38}$

$38 \div 4 = 9 R 2 {(i^{4})}^{9} \cdot i^{2} 1 \cdot - 1$

–1

$i^{67}$

$67 \div 4 = 16 R 3 {(i^{4})}^{16} \cdot i^{3} 1 \cdot - i$

– i

Simplify.

Note

Q: Why must the imaginary unit i be simplified out when the radicand is negative?

A: Because the rules to simplify radicals are only true for real numbers.

$\sqrt{- 18}$

$i \sqrt{3^{2} \cdot 2}$

$3 i \sqrt{2}$

$\sqrt{- 36} \cdot \sqrt{- 16}$

$i \sqrt{6^{2}} \cdot i \sqrt{4^{2}} 6 i \cdot 4 i 24 i^{2}$

–24

Note

Remember to simplify out $\sqrt{- 1}$ before multiplication of any radicands. If your answer is positive, you did not write $\sqrt{- 1}$ as i in your first step.

$\sqrt{- 54}$

$i \sqrt{3^{3} \cdot 2}$

$3 i \sqrt{6}$

$\sqrt{- 32}$

$i \sqrt{2^{5}}$

$4 i \sqrt{2}$

$\sqrt{- 144}$

$i \sqrt{12^{2}}$

12i

$3 \sqrt{- 10} \cdot 7 \sqrt{- 35}$

$3 i \sqrt{2 \cdot 5} \cdot 7 i \sqrt{5 \cdot 7} 21 i^{2} \sqrt{5^{2} \cdot 2 \cdot 7} 105 i^{2} \sqrt{14}$

$- 105 \sqrt{14}$

Note

Q: What is another way to write $i^{2}$ ?

A: Negative one.

$2 \sqrt{- 81} \cdot \sqrt{- 24}$

$2 i \sqrt{9^{2}} \cdot i \sqrt{2^{3} \cdot 3} 18 i \cdot 2 i \sqrt{6} 36 i^{2} \sqrt{6}$

$- 36 \sqrt{6}$

$\sqrt{- 98}$

$i \sqrt{7^{2} \cdot 2}$

$7 i \sqrt{2}$

Simplify. Then classify by all sets to which it belongs: real, pure imaginary, complex.

Note

Q: What classification can you use for every number?

A: Complex

Q: Why is it a good math practice to simplify before you classify numbers?

A: Because the simplified answer may not be the exact classification you originally expected.

$\sqrt{12}$

$2 \sqrt{3}$

Real, complex

$\sqrt{16} + \sqrt{- 8}$

$4 + 2 i \sqrt{2}$

Complex

$\sqrt{- 121}$

11i

Pure imaginary, complex

$5 + \sqrt{49}$

$5 + 7 12$

Real, complex

$3 i$

Pure imaginary, complex

$\sqrt{- 25}$

Pure imaginary, complex

$\sqrt{- 49} \cdot \sqrt{- 4}$

$7 i \cdot 2 i 14 i^{2} 14 (- 1) - 14$

Real, complex

$13 - \sqrt{- 9}$

$13 - 3 i$

Complex