SAT Math - Irrational Numbers

Multiply:

Explanation

Use the FOIL technique:

$= 7 \cdot 1 - 7 \cdot 2i + 3i \cdot 1 - 3i \cdot 2i$

$= 7 - 14i + 3i - 6i ^{2}$

Evaluate:

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

Explanation

We can set in the cube of a binomial pattern:

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

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

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

$= 8 + 36i + 54 i^{2} + 27 i^{3}$

Simplify:

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

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

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

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

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

Simplify:

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.

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

Substitute in for

If $\small x$ and $\small y$ are real numbers, and $\small \small 3x+y+(2x+2y)i=13+10i$ , what is $\small z$ if $\small z=x-4y$ ?

$\small 0$

$\small -15$

$\small -15i$

$\small -13\frac{3}{4}$

$\small -16\frac{1}{4}$

Explanation

To solve for $\small z$ , we must first solve the equation with the complex number for $\small x$ and $\small y$ . We therefore need to match up the real portion of the compex number with the real portions of the expression, and the imaginary portion of the complex number with the imaginary portion of the expression. We therefore obtain:

$\small 3x+y=13$ and $\small 2x+2y=10$

We can use substitution by noticing the first equation can be rewritten as $\small y=13-3x$ and substituting it into the second equation. We can therefore solve for $\small x$ :

$\small 2x+2(13-3x)=10$

$\small 2x+26-6x=10$

$\small -4x=-16$

$\small x=4$

With this $\small x$ value, we can solve for $\small y$ :

$\small 3(4)+y=13$

$\small 12+y=13$

$\small y=1$

Since we now have $\small x$ and $\small y$ , we can solve for $\small z$ :

$\small z=x-4y$

$\small \small z=(4)-4(1)=0$

Our final answer is therefore $\small z=0$

Simplify:

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.

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

Substitute in for .

Evaluate:

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.

The imaginary is equal to:

$\sqrt{-1}$

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

$i=\sqrt{-1}$

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

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

Simplify:

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.

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

Substitute in for

Evaluate $\small \frac{-6+2i}{10-3i}$

$\small \frac{-66+2i}{109}$

You cannot divide by complex numbers

$\small \small \frac{-54+38i}{91-60i}$

$\small \frac{-54+38i}{109}$

$\small \small \frac{-66+2i}{91-60i}$

Explanation

To divide by a complex number, we must transform the expression by multiplying it by the complex conjugate of the denominator over itself. In the problem, $\small 10-3i$ is our denominator, so we will multiply the expression by $\small \frac{10+3i}{10+3i}$ to obtain: