Monomial and Its Standard Form
Definition of Monomial
A monomial is an algebraic expression that contains one term.
Any number, variable, their products and exponents are considered monomials.
For example:
Monomials
Not Monomials
$ 5m^2 n $
$ \left(\frac{3}{4}\right)^2 k $
$8^3$
$ -34m^7 pm^4 z$
abcde
$a^2 b+1$
$ 4(k+n)^2 $
$ 500-m^4+2m^2 $
$ 10p^2+k $
The standard form of a monomial is its expression as the product of one or more factors: a numerical factor (the coefficient) in the first place, and one factor for each variable.
The power of a monomial is the sum of the powers of all the variables.
For example:
$x^2\cdot23xy$ is a monomial in a non-standard form, with the coefficient 23 and the power 4 (x cubed and y y to the $1^{st}$);
$-\frac{3}{15}a^3 b^2$ is a monomial in the standard form, with the coefficient $\left(-\frac{3}{15}\right)$ and the power 5 (a cubed and b squared);
9 is a monomial in the standard form, with the coefficient 9 and the power 0;
a is a monomial in the standard form, with the coefficient 1 and the power 1.
The number 0, as well as monomials identically equal to zero (for example, $0 \cdot x^3, 0\cdot mn$), are called zero-monomials. A zero-monomial is considered to have no exponent.
Monomials with the same literal part (for example, $2ab^3 c^2 and -\frac{7}{5}ab^3 c^2$) are called similar monomials (they are also like terms).
Finding Standard Form of Monomial
Any monomial expression can be transformed into the standard form.
Algorithm of Finding Standard Form of Monomial
- Find the coefficient: multiply all numerical factors and write the result as the first factor.
- Using the properties of exponents, find the total power for each of the variables in the monomial. Write each result as an independent factor.
If a monomial contains several variables, the conventional way is to write the corresponding factors alphabetically. But that is not mandatory.
Examples of Problem Solving
Example 1. Transform the expression into a monomial; find its standard form, coefficient and power:
a) $ \frac{1}{2}x^5y^4c \cdot (-5xy^2 c^3) = \frac{1}{2} \cdot (-5) \cdot c^{1+3} \cdot x^{5+1} \cdot y^{4+2} = -2,5c^4 x^6 y^6 $
coefficient (-2,5), power 4+6+6 = 16
b) $ -(3m^4)^2 \cdot (-m^3 kp)^3 = -3^2 \cdot (-1)^3 \cdot k^3 \cdot m^{8+9} \cdot p^3 = 9k^3 m^17 p^3 $
coefficient 9, power 3+17+3 = 23
c) $ (-2)^3 xy \cdot 1,5(x^4 y)^2 = -8 \cdot 1,5 \cdot x^{1+8} \cdot y^{1+2} = -12x^9 y^3 $
d) $ (8m^3 )^2 n^3 \cdot \frac{1}{(4mn)^3} = \frac{8^2 m^6 n^3}{4^3 m^3 n^3} = \frac{(2^3)^2}{(2^2)^3} \cdot \frac{m^6}{m^3} \cdot \frac{n^3}{n^3} = m^3$
coefficient 1, power 3
Example 2. Find the standard form and evaluate the monomial:
a) $ \frac{1}{2} xy\cdot \frac{1}{4}x^2 for x = 2, y = 3 $
$ \frac{1}{2}xy \cdot \frac{1}{4}x^2 = \frac{1}{2} \cdot \frac{1}{4} \cdot x^{1+2}\cdot y = \frac{1}{8} x^3 y $
Substitute: $ \frac{1}{8}\cdot2^3\cdot3 = 3 $
b) $ (-2a^2 b^3) \cdot \left(\frac{0,5}{ab}\right)^2 for a = 73,b = 3 $
$ (-2a^2 b^3) \cdot \left(\frac{0,5}{ab}\right)^2 = -2 \cdot \frac{1}{2}^2 \cdot \frac{a^2}{a^2} \cdot \frac{b^3}{b^2} = -\frac{1}{2}b $
Substitute: $ -\frac{1}{2}\cdot3 = -1,5 $
Example 3. Write the expression as a square of a monomial:
a) $ 16x^4 y^2 z^6 = 4^2\cdot(x^2 )^2\cdot y^2\cdot(z^3 )^2 = (4x^2 yz^3 )^2 $
b) $ \frac{49}{64}x^{12} y^4 z^{16} = (\frac{7}{8} x^6 y^2 z^8 )^2 $
Example 4*. Given $ 5a^2 b^3 = 7$, evaluate the expression $ -\frac{4}{49} a^6 b^9 $
Evaluate the product: $ a^2 b^3 = \frac{7}{5} $
Transform the expression:
$$ -\frac{4}{49} a^6 b^9 = -\frac{4}{49} \left(\underbrace{a^2 b^3}_{=7/5\text{}}\right)^3 = -\frac{4}{7^2} \cdot \left(\frac{7}{5}\right)^3 = -\frac{4}{5^3} \cdot \frac{7^3}{7^2} = -\frac{28}{125} $$
Answer: $ -\frac{28}{125} $
