Question

name: date: period: score: directions: read each statement carefully. choose and circle the letter of the correct answer. 1.) simplify \\( \sqrt { - 121 } \\) a.) - 11i b.) 11i c.) 11 d.) - 11 2.) simplify \\( i ^ { 25 } \\) a.) -i b.) 1 c.) -1 d.) i 3.) simplify the expression: \\( ( 3 - 2 i ) ( 4 + 5 i ) \\) a.) 2 b.) 22 c.) \\( 2 + 7 i \\) d.) \\( 22 + 7 i \\) 4.) for what values of a and b will the expression \\( ( 7 - 2 i ) ( a + b i ) \\) result in a real number? a = ____ b = ____ 5.) add \\( ( - 3 + 2 i ) + ( 5 - 6 i ) \\) a.) \\( - 2 + 4 i \\) b.) \\( 2 - 3 i \\) c.) \\( - 4 + 2 i \\) d.) \\( 2 - 4 i \\) 6.) subtract \\( ( 11 - 7 i ) - ( - 3 + 12 i ) \\) a.) \\( 14 - 19 i \\) b.) \\( 13 - 19 i \\) c.) \\( 8 + 5 i \\) d.) \\( 8 - 19 i \\) 7.) multiply \\( ( - 5 i ) ( i ) \\) a.) - 5i b.) -5 c.) 5 d.) \\( 5 i ^ { 2 } \\) 8.) simplify \\( 2 i ( 6 - i ) \\) a.) 11i b.) \\( 12 - 2 i \\) c.) \\( 2 + 12 i \\) d.) \\( - 2 + 12 i \\) 9.) multiply \\( ( - 3 + 2 i ) ( 1 - 4 i ) \\) a.) \\( 5 + 14 i \\) b.) \\( - 3 - 8 i \\) c.) \\( - 2 - 2 i \\) d.) \\( - 11 - 10 i \\) 10.) simplify \\( ( 5 - 4 i ) ^ { 2 } \\) a.) \\( 25 + 40 i \\) b.) 41 c.) \\( 9 - 40 i \\) d.) \\( 41 - 40 i \\)

Explanation:

Step1: Simplify \(\sqrt{-121}\)

We know that \(\sqrt{-1}=i\). So, \(\sqrt{-121}=\sqrt{121\times(-1)}=\sqrt{121}\times\sqrt{-1}\). Since \(\sqrt{121} = 11\), then \(\sqrt{-121}=11i\).

Step2: Simplify \(i^{25}\)

We know the pattern of \(i\) powers: \(i^1 = i\), \(i^2=-1\), \(i^3=-i\), \(i^4 = 1\). Divide the exponent \(25\) by \(4\): \(25\div4 = 6\cdots\cdots1\). So \(i^{25}=(i^4)^6\times i^1\). Since \(i^4 = 1\), then \(i^{25}=1^6\times i=i\).

Step3: Simplify \((3 - 2i)(4 + 5i)\)

Use the FOIL method: \((3-2i)(4 + 5i)=3\times4+3\times5i-2i\times4-2i\times5i=12 + 15i-8i-10i^2\). Since \(i^2=-1\), we have \(12 + 7i+10=22 + 7i\).

Step4: For \((7 - 2i)(a+bi)\)

\((7 - 2i)(a+bi)=7a+7bi-2ai-2bi^2=7a + 2b+(7b-2a)i\). For the result to be a real number, the imaginary part \(7b - 2a=0\), or \(a=\frac{7}{2}b\). Let \(b = 2\), then \(a = 7\) (one possible pair).

Step5: Add \((-3 + 2i)+(5-6i)\)

\((-3 + 2i)+(5-6i)=(-3 + 5)+(2i-6i)=2-4i\).

Step6: Subtract \((11-7i)-(-3 + 12i)\)

\((11-7i)-(-3 + 12i)=11-7i + 3-12i=(11 + 3)+(-7i-12i)=14-19i\).

Step7: Multiply \((-5i)(i)\)

\((-5i)(i)=-5i^2\). Since \(i^2=-1\), then \(-5i^2 = 5\).

Step8: Simplify \(2i(6 - i)\)

\(2i(6 - i)=2i\times6-2i\times i=12i-2i^2\). Since \(i^2=-1\), we get \(2 + 12i\).

Step9: Multiply \((-3 + 2i)(1-4i)\)

\((-3 + 2i)(1-4i)=-3\times1+(-3)\times(-4i)+2i\times1+2i\times(-4i)=-3 + 12i+2i-8i^2\). Since \(i^2=-1\), we have \(-3+14i + 8=5+14i\).

Step10: Simplify \((5 - 4i)^2\)

Using \((a - b)^2=a^2-2ab + b^2\), \((5 - 4i)^2=5^2-2\times5\times4i+(4i)^2=25-40i+16i^2\). Since \(i^2=-1\), we get \(25-40i-16=9-40i\).

Answer:

1. b. \(11i\)
2. d. \(i\)
3. d. \(22 + 7i\)
4. \(a = 7\), \(b = 2\) (one possible pair)
5. d. \(2-4i\)
6. a. \(14-19i\)
7. c. \(5\)
8. c. \(2 + 12i\)
9. a. \(5+14i\)
10. c. \(9-40i\)