Number & Quantity - Day 2

Number & Quantity Review - Day 2

Exponents, Imaginary & Complex Numbers, Vectors & Matrices

3. Exponents

Anything to the 0 power is equal to 1.

Exponent Properties

Simplify an exponent of an exponent by multiplying the exponents: (x³)⁴ = x^3×4 = x¹².

Exponents of Exponents

Multiplication and Division

Be careful of common mistakes on exponent problems!

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​Exponent Examples

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Multiple Choice

3x³ · 2x²y · 4x²y is equivalent to:

• 9x⁷y²

• 9x¹²y²

• 24x⁷y²

• 24x¹²y

• 24x¹²y²

Multiple Choice

Which of the following expressions is equivalent to (−2x⁵y²)⁴?

• −16x²⁰y⁸

• −8x²⁰y⁸

• −8x⁹y⁶

• 16x⁹y⁶

• 16x²⁰y⁸

Multiple Choice

Which real number satisfies (2^x)(4) = 8³?

• 2

• 3

• 4

• 4.5

• 7

​4. Imaginary and Complex Numbers

Imaginary Numbers

Any square root of a negaive will result in an imaginary number.

Powers of imaginary Numbers

​To find the value of an imaginary number, use i² = −1.

Ex. (3i)² = (3i)(3i) = (3 × 3)(i × i) = (9)(−1) = −9.

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Remember the following:

i¹ = i

i² = −1

i³ = −i

i⁴ = 1​

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Draw

$$\sqrt[]{-5}$$

(Draw your response in the space provided)

Complex Numbers

A complex number includes the imaginary number i in the form a + bi, such as 3 + 2i. You can represent the complex number as a graph on the number plane, where the horizontal axis, labeled r, represents the real component and the vertical axis, labeled i, represents the imaginary component. For example, to represent the complex number 3 + 2i, plot the point 3 spaces right and 2 spaces up for the coordinates (3,2):

Find the distance, also called the modulus, between two graphed complex numbers with the distance formula or with the pythagorean theorem.

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Find the average or midpoint in the complex plane between two graphed complex numbers with the midpoint formula.​

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Distance & Midpoint

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Multiplication

Multiply or square complex numbers with the FOIL method as you would a quadratic expression:

When a complex number is multiplied by its conjugate, the imaginary component cancels and the real component remains.

Find the conjugate of a complex number by reversing the sign of the imaginary component.

(3 + 2i)²

5. Vectors

A vector is an object, represented by an arrow, that has both magnitude and direction. It’s written in the component form as a,b, where, from any starting point, a represents the number of units that it moves right and b represents the number of units that it moves up. In this way, the component form represents changes in its x and y-values.

​For example, the vector〈4,3〉moves 4 units right and 3 units up:

Magnitude of a vector

Think of the vector as a right triangle, with its component form as the sides and its magnitude as the hypotenuse. The component form can have negative values, meaning the vector moves left and/or down. However, the magnitude is always positive:

The distance between the starting point and ending point is the magnitude, which can be found using the Pythagorean theorem of its component form. Place the component numbers in the theorem as a and b, with c as the magnitude:

The vector can be multiplied by a number, known as a scalar. For example, to multiply the previous vector by a scalar of 2, write it as 2〈4,3〉 =〈8,6〉, in which

case the vector moves 8 units right and 6 units up. Find the magnitude by placing these component numbers into the Pythagorean theorem:

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​vector multiplication

Vector addition and Subtraction

To add or subtract vectors, use their component forms to separately add or subtract the x-values and y-values.

Ex: 〈2,3〉 + 〈5,1〉 = 〈7,4〉

In other words, if vector a moves right 2 and up 3, and vector b moves right 5 and up 1, and a + b = c, then vector c moves right 2 + 5 = 7 and up 3 + 1 = 4:​

Note that the resulting magnitude isn’t the sum of the two starting magnitudes. It’s calculated from the component form of the new vector with the Pythagorean theorem.

Multiple Choice

The component forms of vectors u and v are given by u = 〈5,3〉 and v = 〈2, −7〉. Given that 2u + (−3v) + w = 0, what is the component form of w?

• 〈−16,15〉

• 〈−4, −27〉

• 〈3,10〉

• 〈4,27〉

• 〈16, −15〉

6. Matrices

A matrix is a rectangular array of letters or numbers that represents data. The matrix facilitates data manipulation, making it ideal for use in complex applications such as statistics or computer operations.

For example, is a 2 × 3 matrix, because it has two rows and three columns.

number of rows x number of columns

Matrix Addition and Subtraction

Matrices can only be added or subtracted when they have the same dimensions. For example, a 3 × 2 matrix can only be added or subtracted to another 3 × 2 matrix. To do this, add (or subtract) the quantities in each corresponding position.​

Subject | Subject

Some text here about the topic of discussion

​+

​=

Matrix Multiplication

A matrix can be multiplied by a single quantity, also known as a scalar. To do this, distribute the scalar among the values in the matrix, and the result is a scalar multiple.

This is just like scalar multiplication of vectors.

​=

​Practice with Matrices

​Practice with Matrices

​=

​=

​=

Matrix x Matrix

A matrix can be multiplied by another matrix when the number of columns in the first matrix matches the number of rows in the second matrix.

Subject | Subject

Some text here about the topic of discussion

​=

Multiple Choice

Given these matrices, what is the athletic director's estimate for the number of sports awards that will be earned for these fall sports?

• 80

• 88

• 91

• 92

• 99

Matrix Determinant

The determinant that appears in the ACT mathematics test is based on a 2 × 2 matrix, where the determinant of is ad − bc.

If the matrix has numbers, find the determinant by placing the numbers into the equation.

​For example, the determinant of

Fill in the Blanks

Fill in the Blanks

Multiple Choice

• -138

• -42

• 12

• 42

• 138