# Important Numbers and Operations Formulas for the ISEE Math Tests

Taking the ISEE Math tests can be confusing if you don’t know the difference between the two of them. The Mathematics Achievement test focuses on calculations, while the Quantitative Reasoning test focuses on reasoning. But they have something in common: For both of the tests you’ll need to know how to think mathematically and how to handle math expressions.

There are four main areas of math covered in these two tests and you’ll need to know the most important formulas for each one of them. In the following chart, you’ll find formulas for Numbers and Operations. Also, be sure to check our other three FREE formula charts so you can ace the ISEE Math Tests:

For more information, feel free to check out our FREE study guides, practice questions, and flashcards.

## Numbers and Operations Formulas for the ISEE Math Tests

Category Formula Symbols Comment
Numbers and Operations $$a+b=b+a$$

$$a \cdot b=b \cdot a$$
a, b = any constant or variable Commutative Property
Numbers and Operations $$a+(b+c)=(a+b)+c$$

$$a \cdot (b \cdot c)=(a \cdot b) \cdot c$$
a, b, c = any constant or variable Associative Property
Numbers and Operations $$a \cdot (b+c)=a \cdot b+a \cdot c$$ a, b, c = any constant or variable Distributive Property
Numbers and Operations $$a + 0 = a$$ a = any constant or variable Identity Property of Addition
Numbers and Operations $$a \cdot 1 = a$$ a = any constant or variable Identity Property of Multiplication
Numbers and Operations $$\frac{a}{b} + \frac{c}{d} = \frac {(a \cdot d)+(b \cdot c)}{b \cdot d}$$ a, b, c, d = any real number Remember to simplify the fraction if possible.
Numbers and Operations $$\frac{a}{b} \cdot \frac{c}{d} = \frac {(a \cdot c)}{b \cdot d}$$ a, b, c, d = any real number Remember to simplify the fraction if possible.
Numbers and Operations $$\frac{a}{b} \div \frac{c}{d} = \frac {(a \cdot d)}{b \cdot c}$$ a, b, c, d = any real number Remember to simplify the fraction if possible.
Numbers and Operations $$a\frac{b}{c} = \frac{(a \cdot c) + b}{c}$$ a, b, c = any real number Remember to simplify the fraction if possible.
Numbers and Operations $$x^a \cdot x^b = x^{a+b}$$ a, b, x = any real number
Numbers and Operations $$\frac{x^a}{x^b} = x^{a-b}$$ a, b, x = any real number
Numbers and Operations $$(x^a)^b = x^{a \cdot b}$$ a, b, x = any real number
Numbers and Operations $$(x \cdot y)^a = x^a \cdot y^a$$ a, b, x, y = any real number
Numbers and Operations $$x^1=x$$ x = any real number
Numbers and Operations $$x^0=1$$ x = any real number
Numbers and Operations $$x^{-a} = \frac{1}{x^a}$$ a, x = any real number
Numbers and Operations $$a \cdot b\% = a \cdot \frac{b}{100}$$ a = any real number
b% = any percent
Remember to simplify if possible
Numbers and Operations $$\% = \frac {\vert b-a \vert}{b} \cdot 100 = \frac{c}{b} \cdot 100$$ % = % increase or decrease
a = new value
b = original value
c = amount of change

Numbers and Operations $$if(ad−bc\neq0),\begin{pmatrix} a&b \\ c &d \end{pmatrix}^{−1}=\frac{1}{ad−bc}\begin{pmatrix} d&−b \\ −c &a \end{pmatrix}$$ a, b, c, d = any real number Inverse of a matrix
Numbers and Operations $$\begin{vmatrix}a_{11} &a_{12} \\a_{21}&a_{22} \\ \end{vmatrix}= a_{11} \cdot a_{22} - a_{12} \cdot a_{21}$$ $$a_{nn}$$ = any real number Determinant of a 2 x 2 matrix
Numbers and Operations $$det\begin{pmatrix}a_{11} &a_{12} &a_{13} \\a_{21}&a_{22} &a_{23} \\ a_{31} &a_{32} &a_{33}\end{pmatrix}$$

$$=a_{11}\begin{vmatrix}a_{22}&a_{23} \\ a_{32} & a_{33}\end{vmatrix}+a_{12}\begin{vmatrix} a_{21}&a_{23} \\a_{31} & a_{33}\end{vmatrix}+a_{13}\begin{vmatrix}a_{21}&a_{22} \\ a_{31} & a_{32}\end{vmatrix}$$
$$a_{nn}$$ = any real number Determinant of a 3 x 3 matrix
Numbers and Operations $$\vert \vert a \vert \vert = \sqrt{a_1^2 + a_2^2 + a_3^2}$$ Vector a = $$(a_1, a_2, a_3)$$ Length of a vector
Numbers and Operations $$a \cdot b = a_1b_1 + a_2b_2 + a_3b_3$$ Vector a = $$(a_1, a_2, a_3)$$
Vector b = $$(b_1, b_2, b_3)$$
Dot product of vectors
Numbers and Operations $$\theta = \cos^{-1} (\frac{a \cdot b}{\vert \vert a \vert \vert \cdot \vert \vert b \vert \vert})$$
$$\theta$$ = angle between vectors
Vector a = $$(a_1, a_2, a_3)$$
Vector b = $$(b_1, b_2, b_3)$$
Angle between vectors
Numbers and Operations $$a\times b=\begin{vmatrix}\hat{i} &\hat{j} &\hat{k} \\ a_1&a_2 &a_3 \\ b_1 &b_2 &b_3 \end{vmatrix}$$ Vector a = $$(a_1, a_2, a_3)$$

Vector b = $$(b_1, b_2, b_3)$$
Cross product of vectors