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  \(\frac{x^a}{x^b} = x^{ab}\)  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  \(\% = \frac {\vert ba \vert}{b} \cdot 100 = \frac{c}{b} \cdot 100\)  % = % 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 
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