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PSAT Math : Algebra

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Example Questions

Example Question #2 : Complex Numbers

Define an operation $\ast$ as follows:

For all complex numbers a, b ,

$a \ast b = \frac{ab}{i}$ .

Evaluate $(3+ 4i) \ast (4-3i)$ .

Possible Answers:

7 +24i

25i

7 - 24i

-25i

24-7i

Correct answer:

7 - 24i

Explanation:

$a \ast b = \frac{ab}{i}$

$(3+ 4i) \ast (4-3i)= \frac{(3+ 4i) (4-3i)}{i}$

$= \frac{ 3 \cdot 4 - 3 \cdot 3i + 4i \cdot 4 -4i \cdot 3i}{i}$

$= \frac{ 12 - 9i + 16i -12i ^{2}}{i}$

$= \frac{ 12 - 9i + 16i -12 (-1)}{i}$

$= \frac{ 12 - 9i + 16i +12}{i}$

$= \frac{ 24+7i }{i}$

$= \frac{ \left (24+7i \right ) (-i)}{i (-i)}$

$= \frac{ -24i-7i ^{2} }{ -i^{2}}$

$= \frac{ -24i-7(-1) }{ -(-1)}}$

$= \frac{ -24i+7 }{ 1}$

= 7 - 24i

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Example Question #4 : How To Divide Complex Numbers

Define an operation $\ast$ as follows:

For all complex numbers a, b ,

$a \ast b = \frac{ a^{3}}{b^{3} + 1}$

Evaluate $i \ast i$ .

Possible Answers:

1-i

$\frac{ 1}{2 }i$

$\frac{1}{2}+ \frac{ 1}{2 }i$

$- \frac{ 1}{2 }i$

$\frac{1}{2}- \frac{ 1}{2 }i$

Correct answer:

$\frac{1}{2}- \frac{ 1}{2 }i$

Explanation:

$a \ast b = \frac{ a^{3}}{b^{3} + 1}$

$i \ast i = \frac{ i^{3}}{i^{3} + 1}$

$= \frac{ -i}{-i+ 1}$

$= \frac{ -i}{1-i }$

$= \frac{ i}{-1+i }$

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is -1 - i . The above becomes:

$\frac{ i\left (-1-i \right ) }{\left (-1+i \right )\left (-1-i \right ) }$

$= \frac{ -i-i^{2} }{ 1^2+i-i-i^2 }$

$=\frac{-i-i^2}{1-i^2}$

$= \frac{ -i-(-1) }{ 1+ 1 }$

$= \frac{ -i+1 }{2 }$

$= \frac{ 1-i }{2 }$

$= \frac{1}{2}- \frac{ 1}{2 }i$

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Example Question #521 : Algebra

Which of the following is equal to $\left (3i \right )^{5}$ ?

Possible Answers:

-243i

15i

-15i

243i

None of the other responses is correct.

Correct answer:

243i

Explanation:

By the power of a product property,

$\left (3i \right )^{5} = 3^{5} \cdot i^{5} = 3^{5} \cdot i^{4} \cdot i = 243 \cdot 1 \cdot i = 243 i$

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Example Question #1 : How To Multiply Complex Numbers

Multiply:

(10 + 3i)(10-3i)

Possible Answers:

100 + 9i

109

100 - 9i

None of the other responses is correct.

Correct answer:

109

Explanation:

This is the product of a complex number and its complex conjugate. They can be multiplied using the pattern

$\left (A + Bi \right )\left (A - Bi \right ) = A^{2} + B^{2}$

with A = 10, B = 3

$(10 + 3i)(10-3i) = 10^{2} +3^{2} = 100 + 9 = 109$

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Example Question #11 : Complex Numbers

Which of the following is equal to $i ^{-56}$ ?

Possible Answers:

Correct answer:

Explanation:

$i ^{-56} = \frac{1}{i^{56}}$

$56 \div 4 = 14$ , so 56 is a multiple of 4. raised to the power of any multiple of 4 is equal to 1, so

$i ^{-56} = \frac{1}{i^{56}} = \frac{1}{1} = 1$ .

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Example Question #1 : How To Multiply Complex Numbers

Which of the following is equal to (2i)^7 ?

Possible Answers:

-2i

-128i

128i

Correct answer:

-128i

Explanation:

The first step to solving this problem is distributing the exponent:

2^7i^7 = 128i^7

Next, we need simplify the complex portion.

$i^7=i^2\times i^2\times i^2\times i=-1\times -1\times -1\times i=-i$

Thus, our final answer is $128\times (-i)=-128i$ .

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Example Question #2 : How To Multiply Complex Numbers

What is the eighth power of ?

Possible Answers:

6,561i

None of the other responses is correct.

-24i

-6,561i

24i

Correct answer:

None of the other responses is correct.

Explanation:

$\left (3i \right )^{8}$

$= 3^{8} i^{8}$

$= 6,561 i^{8}$

raised to the power of any multiple of 4 is equal to 1, so the above expresion is equal to

$= 6,561 \cdot 1 = 6,561$

This is not among the given choices.

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Example Question #4 : How To Multiply Complex Numbers

What is the third power of $2 + i\sqrt{5}$ ?

Possible Answers:

$38 +7i\sqrt{5}$

$-22 +17i\sqrt{5}$

$-22 +7i\sqrt{5}$

38 +35i

-22 +35i

Correct answer:

$-22 +7i\sqrt{5}$

Explanation:

You are being asked to evaluate

$\left (2 + i\sqrt{5} \right )^{3}$

You can use the cube of a binomial pattern with $A = 2, B = i\sqrt{5}$ :

$\left ( A + B \right )^{3} = A^{3}+ 3A^{2}B + 3AB^{2}+ B^{3}$

$\left ( 2 + i\sqrt{5} \right )^{3} =2^{3}+ 3 \cdot 2^{2} \cdot i\sqrt{5} + 3 \cdot 2 (i\sqrt{5}) ^{2}+ (i\sqrt{5}) ^{3}$

$=8+ 3 \cdot 4 \cdot i\sqrt{5} + 6i ^{2} (\sqrt{5}) ^{2}+ i^{3}(\sqrt{5}) ^{3}$

$=8+12i\sqrt{5} + 6 (-1) 5+(-i)(5\sqrt{5})$

$=8-30 +12i\sqrt{5} -5i\sqrt{5}$

$=-22 +7i\sqrt{5}$

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Example Question #5 : How To Multiply Complex Numbers

What is the fourth power of $3 - i\sqrt{2}$ ?

Possible Answers:

$121-84 i\sqrt{2}}$

$-23-84 i\sqrt{2}}$

$49- 132 i\sqrt{2}$

-23-168i

95-168i

Correct answer:

$-23-84 i\sqrt{2}}$

Explanation:

$\left (3 - i\sqrt{2} \right )^{4}$ can be calculated by squaring $3 - i\sqrt{2}$ , then squaring the result, using the square of a binomial pattern as follows:

$\left (3 - i\sqrt{2} \right )^{2}$

$=3^{2}- 2 \cdot 3 \cdot i\sqrt{2} +\left ( i\sqrt{2} \right )^{2}$

$=9- 6 i\sqrt{2} +2i^{2}$

$=9- 6 i\sqrt{2} +2(-1)$

$=7- 6 i\sqrt{2}$

$\left (3 - i\sqrt{2} \right )^{4}$

$=\left [ \left (3 - i\sqrt{2} \right )^{2} \right ]^{2}$

$=\left (7- 6 i\sqrt{2}} \right )^{2}$

$=7^{2} - 2 \cdot 7 \cdot 6 i\sqrt{2}} + \left ( 6 i\sqrt{2}} \right )^{2}$

$=49-84 i\sqrt{2}} + 6^{2} i^{2} \left ( \sqrt{2}} \right )^{2}$

$=49-84 i\sqrt{2}} +36 (-1) (2)$

$=49-84 i\sqrt{2}} -72$

$=-23-84 i\sqrt{2}}$

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Example Question #13 : Complex Numbers

Multiply 32+ 18i by its complex conjugate. What is the product?

Possible Answers:

1,152

700 + 1,152i

700

700 - 1,152i

1,348

Correct answer:

1,348

Explanation:

The product of any complex number a + bi and its complex conjugate a - bi is the real number $a^{2}+b^{2}$ , so all that is needed here is to evaluate the expression:

$32^{2}+ 18 ^{2}= 1,024+ 324 = 1,348$

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PSAT Math : Algebra

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #2 : Complex Numbers

Example Question #4 : How To Divide Complex Numbers

Example Question #521 : Algebra

Example Question #1 : How To Multiply Complex Numbers

Example Question #11 : Complex Numbers

Example Question #1 : How To Multiply Complex Numbers

Example Question #2 : How To Multiply Complex Numbers

Example Question #4 : How To Multiply Complex Numbers

Example Question #5 : How To Multiply Complex Numbers

Example Question #13 : Complex Numbers

All PSAT Math Resources

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