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SAT II Math I : Irrational Numbers

Study concepts, example questions & explanations for SAT II Math I

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All SAT II Math I Resources

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Example Questions

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Example Question #11 : Number Theory

Simplify:

(2+5i)(-8-i)

Possible Answers:

42+11i

-11-42i

-18-5i

12+6i

Correct answer:

-11-42i

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(2+5i)(-8-i)=-16-2i-40i-5i^2

Remember that $i=\sqrt-1$ , so i^2=-1 .

Substitute in for i^2

-16-2i-40i-5i^2=-16-42i-5(-1)=-16+5-42i=-11-42i

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Example Question #12 : Number Theory

Simplify:

(2+6i)(5+3i)

Possible Answers:

36-8i

-36-8i

12-8i

-8+36i

Correct answer:

-8+36i

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(2+6i)(5+3i)=10+6i+30i+18i^2

Remember that $i=\sqrt-1$ , so i^2=-1 .

Substitute in for i^2

10+6i+30i+18i^2=10+36i+18(-1)=10-18+36i=-8+36i

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Example Question #13 : Number Theory

Simplify:

(2+5i)(2-5i)

Possible Answers:

16-5i

4-10i

Correct answer:

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(2+5i)(2-5i)=4-10i+10i-25i^2

Remember that $i=\sqrt{-1}$ , so i^2=-1 .

Substitute in for i^2 .

4-10i+10i-25i^2=4-25(-1)=4+25=29

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Example Question #14 : Number Theory

Simplify:

$\frac{8-3i}{2-i}$

Possible Answers:

$\frac{7+2i}{3}$

3i+6

10-7i

$\frac{19+2i}{5}$

Correct answer:

$\frac{19+2i}{5}$

Explanation:

To get rid of the fraction, multiply the numerator and denominator by the conjugate of the denominator.

$\frac{8-3i}{2-i}\left(\frac{2+i}{2+i}\right)$

Now, multiply and simplify.

$\frac{8-3i}{2-i}\left(\frac{2+i}{2+i}\right)=\frac{16+8i-6i-3i^2}{4-i^2}$

Remember that i^2=-1

$\frac{16+8i-6i-3i^2}{4-i^2}=\frac{16+2i-3(-1)}{4-(-1)}=\frac{19+2i}{5}$

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Example Question #15 : Number Theory

Write in standard form: $\frac{3}{2+4i}$

Possible Answers:

$\frac{6-12i}{20}$

6-12i

$\frac{2+4i}{3}$

None of the other answers

$\frac{3}{10}-\frac{3i}{5}$

Correct answer:

$\frac{3}{10}-\frac{3i}{5}$

Explanation:

$\frac{3}{2+4i}$

Multiply by the conjugate:

$\frac{3}{2+4i}*\frac{2-4i}{2-4i}$

$\frac{3(2-4i)}{4-16i^2}$

$\frac{6-12i}{4-16(-1)}$

$\frac{6-12i}{20}=\frac{6}{20}-\frac{12i}{20}=\mathbf{\frac{3}{10}-\frac{3i}{5}}$

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Example Question #1 : Understanding Imaginary And Complex Numbers

Simplify the expression.

4i^2-6i-7i^2+3i+4

Possible Answers:

None of the other answer choices are correct.

-3i^2-3i+4

7-3i

-7-3i

Correct answer:

7-3i

Explanation:

Combine like terms. Treat $\small i$ as if it were any other variable.

4i^2-6i-7i^2+3i+4

-3i^2-3i+4

Substitute to eliminate $\small i^2$ .

i^2=-1

-3(-1)-3i+4

Simplify.

3-3i+4=7-3i

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Example Question #4 : Basic Operations With Complex Numbers

What is the value of (3 + i)(4 + i) ?

Possible Answers:

12 + 6i

11 + 7i

12 + 8i

Correct answer:

11 + 7i

Explanation:

When dealing with imaginary numbers, we multiply by foiling as we do with binomials. When we do this we get the expression below:

(3+i)(4+i)= 12 + 3i + 4i + i^2

Since we know that i^2 = -1 we get 12 + 7i - 1 which gives us 11 + 7i .

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Example Question #21 : Sat Subject Test In Math I

$\frac{1-7i}{6-2i}=a-i$

Find .

Possible Answers:

2.5

0.5

Correct answer:

0.5

Explanation:

Multiply the numerator and denominator by the numerator's complex conjugate.

$\frac{1-7i}{6-2i}\ast \frac{6+2i}{6+2i}=\frac{20-40i}{40}$

Reduce/simplify.

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Example Question #61 : Imaginary Numbers

Multiply:

(2+3i)(1-i)

Answer must be in standard form.

Possible Answers:

5-i

5+i

2-3i

2+2i

Correct answer:

5+i

Explanation:

(2+3i)(1-i)

The first step is to distribute which gives us:

$2-2i+3i-3i^{2}$

$2+i-3i^{2}=2+i+3=5+i$

which is in standard form.

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Example Question #1 : Complex Numbers

Evaluate: i(1+i)(2-i)

Possible Answers:

1-3i

1+3 i

-1-3i

-3+i

-1+3i

Correct answer:

-1+3i

Explanation:

Use the FOIL method to simplify. FOIL means to mulitply the first terms together, then multiply the outer terms together, then multiply the inner terms togethers, and lastly, mulitply the last terms together.

i(1+i)(2-i)

=i[(1)(2)+(1)(-i)+(i)(2)+(i)(-i)]

=i[2-i+2i-i^2]

=i[2+i-i^2]

=2i+i^2-i^3

The imaginary is equal to:

$\sqrt{-1}$

Write the terms for $i, i^2, \textup{and } i^3$ .

$i=\sqrt{-1}$

i^2=-1

$i^3=i \times i^2=-\sqrt{-1}=-i$

Replace $i^2 \textup{ and } i^3$ with the appropiate values and simplify.

2i+i^2-i^3= 2i-1-(-i) = -1+3i

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SAT II Math I : Irrational Numbers

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #11 : Number Theory

Example Question #12 : Number Theory

Example Question #13 : Number Theory

Example Question #14 : Number Theory

Example Question #15 : Number Theory

Example Question #1 : Understanding Imaginary And Complex Numbers

Example Question #4 : Basic Operations With Complex Numbers

Example Question #21 : Sat Subject Test In Math I

Example Question #61 : Imaginary Numbers

Example Question #1 : Complex Numbers

All SAT II Math I Resources

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