Real and Complex Numbers - SAT Subject Test in Math II

Card 0 of 19

Question

Evaluate:

$\left ( 7+3i\right )^{2}$

Answer

Use the square of a sum pattern

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

where :

$\left ( 7+3i\right )^{2} = 7^{2}+ 2 \cdot 7 \cdot 3i + (3i)^{2}$

$= 7^{2}+ 2 \cdot 7 \cdot 3i + 3^{2}i^{2}$

Compare your answer with the correct one above

Question

Multiply: $\left ( 4+ 7i\right )(4 - 7i)$

Answer

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

$\left (4 + 7i \right )\left (4 - 7i \right ) = 4^{2} + 7^{2} = 16 + 49 = 65$

This is not among the given responses.

Compare your answer with the correct one above

Question

Multiply:

$-3 i \cdot 8i \cdot 5i \cdot 6i$

Answer

$-3 i \cdot 8i \cdot 5i \cdot 6i$

$= \left (-3 \cdot 8 \cdot 5 \cdot 6 \right ) \cdot\left ( i \cdot i \cdot i \cdot i \right )$

$= \left (-720 \right ) \cdot\left ( i ^{4}\right )$

$-720 \cdot 1 = -720$

Compare your answer with the correct one above

Question

Which of the following is equal to $i^{37}$ ?

Answer

To raise to a power, divide the exponent by 4 and note the remainder.

$37 \div 4 = 9 \textrm{ R }1$

Raise to the power of that remainder:

$i^{37} = i^{1} = i$

Compare your answer with the correct one above

Question

Evaluate:

$(6 +5i)^{2}$

Answer

Use the square of a sum pattern

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

where :

$(6 +5i)^{2} = 6^{2} + 2 \cdot 6i \cdot 5 + (5i)^{2}$

$= 6^{2} + 2 \cdot 6 \cdot 5i + 5^{2}i^{2}$

Compare your answer with the correct one above

Question

Multiply:

Answer

Apply the distributive property:

$= 7i \cdot 5 + 7i \cdot 9i$

$= 35i + 63 i^{2}$

Compare your answer with the correct one above

Question

Which of the following is equal to $\left ( 4i\right )^{4}$ ?

Answer

By the power of a product property,

$\left ( 4i\right )^{4} = 4^{4} \cdot i^{4} = 256 \cdot 1 = 256$

Compare your answer with the correct one above

Question

Multiply:

Answer

Use the FOIL method:

$= 6 \cdot 7 +6 \cdot 6i - 7 \cdot 7i - 7i \cdot 6i$

$=42 +36i - 49i - 42i ^{2}$

Compare your answer with the correct one above

Question

Which of the following is equal to $i^{77}$ ?

Answer

To raise to a power, divide the exponent by 4 and note the remainder.

$77 \div 4 = 19\textrm{ R }1$

Raise to the power of that remainder:

$i^{77} = i^{1} = i$

Compare your answer with the correct one above

Question

Multiply:

Answer

Apply the distributive property:

$= 8i \cdot 5- 8i \cdot 7i$

$= 40i- 56i^{2}$

Compare your answer with the correct one above

Question

Multiply:

$\left ( -4i\right ) (-6i)(-7i)$

Answer

$\left ( -4i\right ) (-6i)(-7i)$

$= \left ( -4\right ) (-6)(-7) \cdot i \cdot i \cdot i$

$= -168 \cdot i ^{3} = -168 (-i) = 168i$

Compare your answer with the correct one above

Question

$\overline{x}$ denotes the complex conjugate of .

If $x = \frac{1}{10} - \frac{1}{5} i$ , then evaluate $\overline{x}^{2} - x^{2}$ .

Answer

By the difference of squares pattern,

$\overline{x}^{2} - x^{2} = (\overline{x}+ x) ( \overline{x} - x)$

If $x = \frac{1}{10} - \frac{1}{5} i$ , then $\overline{x} = \frac{1}{10} + \frac{1}{5} i$ .

Consequently:

$\overline{x} + x =\left ( \frac{1}{10} + \frac{1}{5} i \right ) + \left ( \frac{1}{10} - \frac{1}{5} i \right )$

$= \frac{1}{10}+ \frac{1}{10} + \frac{1}{5} i- \frac{1}{5} i$

$= \frac{1}{5}$

$\overline{x} - x =\left ( \frac{1}{10} + \frac{1}{5} i \right ) - \left ( \frac{1}{10} - \frac{1}{5} i \right )$

$= \frac{1}{10}- \frac{1}{10} + \frac{1}{5} i+ \frac{1}{5} i$

$= \frac{2}{5} i$

Therefore,

$\overline{x}^{2} - x^{2} = (\overline{x}+ x) ( \overline{x} - x) = \frac{1}{5} \cdot \frac{2}{5} i = \frac{2}{25} i$

Compare your answer with the correct one above

Question

is a complex number; $\overline{x}$ denotes the complex conjugate of .

$x \cdot \overline{x} = 84$

Which of the following could be the value of ?

Answer

The product of a complex number and its complex conjugate $\overline{x} = a- bi$ is

$x \cdot \overline{x} = (a+ bi)(a-bi) = a^{2} + b^{2}$

Setting and accordingly for each of the four choices, we want to find the choice for which $x \cdot \overline{x} = a^{2} + b^{2} = 84$ :

$6 - 4 i \sqrt{3}$

$a = 6, b = - 4 \sqrt{3}$

$x \cdot \overline{x} = a^{2} + b^{2} = 6^{2}+ (-4 \sqrt{3})^{2} = 36 + 4 ^{2} (\sqrt{3} )^{2} = 36 + 16 \cdot 3 = 36 + 48 = 84$

$7 + i \sqrt{35}$

$a = 7, b = \sqrt{35}$

$x \cdot \overline{x} = a^{2} + b^{2} = 7^{2} + (\sqrt{35})^{2} = 49 + 35 = 84$

$8 + 2 i \sqrt{5}$

$a = 8 , b = 2 \sqrt{5}$

$x \cdot \overline{x} = a^{2} + b^{2} = 8^{2} + (2 \sqrt{5})^{2} = 64 + 2^{2} \cdot (\sqrt{5})^{2} = 64 + 4 \cdot 5 = 64 + 20 = 84$

$9 - i \sqrt{3}$

$a = 9 , b = - \sqrt{3}$

$x \cdot \overline{x} = a^{2} + b^{2} = 9^{2} + (- \sqrt{3})^{2} = 81 + 3 = 84$

For each given value of , $x \cdot \overline{x} = 84$ .

Compare your answer with the correct one above

Question

$\overline{x}$ denotes the complex conjugate of .

If $x = 5 + i \sqrt{2}$ , then evaluate $x^{4} \cdot \overline{x} ^{4}$ .

Answer

Applying the Power of a Product Rule:

$x^{4} \cdot \overline{x} ^{4}=(x \overline{x})^{4}$

The complex conjugate of an imaginary number is $\overline{x} = a - bi$ ; the product of the two is

$x \overline{x} = a^{2} + b^{2}$

$x = 5 + i \sqrt{2}$ , so, setting $a = 5 , b = \sqrt{2}$ in the above pattern:

$x \overline{x} = 5^{2} + (\sqrt{2})^{2} = 25 + 2= 27$

Consequently,

$x^{4} \cdot \overline{x} ^{4}=(x \overline{x})^{4} = 27 ^{4} = 531, 441$

Compare your answer with the correct one above

Question

Let and be complex numbers. $\overline{x}$ and $\overline{y}$ denote their complex conjugates.

$\overline{x} \cdot \overline{y } = 7 + 8 i \sqrt{2}$

$y = 8 - i\sqrt{3}$

Evaluate .

Answer

Knowing the actual values of and is not necessary to solve this problem. The product of the complex conjugates of two numbers is equal to the complex conjugate of the product of the numbers; that is,

$\overline{xy }= \overline{x} \cdot \overline{y }$

We are given that $\overline{xy }= \overline{x} \cdot \overline{y } = 7 + 8 i \sqrt{2}$ . is therefore the conjugate of $\overline{xy }$ , or $7 - 8 i \sqrt{2}$ .

Compare your answer with the correct one above

Question

Let be a complex number. $\overline{x}$ denotes the complex conjugate of .

$x+\overline{ x} = 20$ and $x \overline{ x} = 200$ .

How many of the following expressions could be equal to ?

(a)

(b)

(c)

(d)

Answer

is a complex number, so for some real ; also, $\overline{x} = a-bi$ .

Therefore,

$x+\overline{ x} = (a+ bi) + (a-bi) = a+ a + bi - bi = 2a$

Substituting:

$x+\overline{ x} = 20$

Therefore, we can eliminate choices (c) and (d).

Also, the product

$x \overline{ x} = (a+bi)(a-bi) = a^{2} +b^{2}$

Setting $x \overline{ x} = 200$ and substituting 10 for , we get

$10^{2} +b^{2} = 200$

$100 +b^{2} = 200$

$b^{2} = 100$

$b = \pm 10$

Therefore, either or - making two the correct response.

Compare your answer with the correct one above

Question

Evaluate $i^{99}$

Answer

To evaluate a power of , divide the exponent by 4 and note the remainder.

$99 \div 4 = 24\textrm{ R }3$

The remainder is 3, so

$i^{99} = i^{3}$

Consequently, using the Product of Powers Rule:

$i^{99} = i^{2} \cdot i = -1 \cdot i = -i$

Compare your answer with the correct one above

Question

The fraction $\frac{1+i}{1-i}$ is equivalent to which of the following?

Answer

Start by multiplying the fraction by $\frac{1+i}{1+i}$ .

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

Since , we can then simplify the fraction:

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

Thus, the fraction is equivalent to .

Compare your answer with the correct one above

Question

The fraction $\frac{5+3i}{2-2i}$ is equivalent to which of the following?

Answer

Start by multiplying both the denominator and the numerator by the conjugate of , which is .

$\frac{5+3i}{2-2i}(\frac{2+2i}{2+2i})=\frac{10+10i+6i+6i^2}{4-4i^2}$

Next, recall , and combine like terms.

$\frac{10+10i+6i+6i^2}{4-4i^2}=\frac{10+16i+6(-1)}{4-4(-1)}=\frac{4+16i}{8}$

Finally, simplify the fraction.

$\frac{4+16i}{8}=\frac{1}{2}+2i$

Compare your answer with the correct one above

Tap the card to reveal the answer