Multiplication and Exponentiation of Monomials
Algorithm for Multiplication of Monomials
- Find the coefficient of the resulting monomial: multiply all the coefficients of the multiplied monomials.
- Using the properties of exponents and multiplication, find the total power for each of the variables. Write the corresponding factors.
The product of monomials is a monomial.
Consider: $ (\frac{2}{3} a^2 b) \cdot (3ab^5 )$
Step 1. Coefficient $ \frac{2}{3} \cdot 3 = 2 $
Step 2. Powers of the variables: $ a^{2+1} = a^3, b^{1+5} = b^6 $
Therefore: $(\frac{2}{3} a^2 b) \cdot (3ab^5 ) = \frac{2}{3} \cdot 3\cdot a^{2+1} \cdot b^{1+5} = 2a^3 b^6$
Algorithm for Exponentiation Of Monomials
Raise each monomial factor to the power and multiply the results.
A monomial raised to a power is a monomial.
Consider: $ (\frac{2}{3} a^2 b^5)^3 $
Raise each factor to the power: $ \left(-\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125}, (a^2 )^3 = a^6,(b^5 )^3 = b^15 $
Therefore: $ (\frac{2}{3} a^2 b^5)^3 = \left(-\frac{2}{5}\right)^3 \cdot (a^2 )^3 \cdot (b^5 )^3 = \frac{8}{125} a^6 b^{15} $
Examples of Problem Solving
Example 1. Multiply the monomials:
a) $ (-3a^2 xy^5 )\cdot( \frac{1}{9} ax^4 y^2 ) = -3 \cdot \frac{1}{9} \cdot a^{2+1} \cdot x^{1+4} \cdot y^{5+2} = -\frac{1}{3} a^3 x^5 y^7 $
b) $ (7az)\cdot(- \frac{1}{4} a^2 xy) \cdot (16xz^4 ) = -7\cdot \frac{1}{4} \cdot 16 \cdot a^{1+2}\cdot x^{1+1}\cdot z^{1+4} = -28a^3 x^2 z^5 $
Example 2. Find the cube of the monomial:
a) $ (3a^5 by^2 )^3 = 3^3\cdot(a^5 )^3\cdot b^3\cdot(y^2 )^3 = 27a^{15} b^3 y^6 $
b) $ \left(-\frac{1}{2}xy^2 z^7\right)^3 = \left(-\frac{1}{2}\right)^3\cdot x^3\cdot(y^2 )^3\cdot(z^7 )^3 = -\frac{1}{8} x^3 y^6 z^{21} $
Example 3. Simplify the expression:
a) $ 3x^5 \cdot \left(\frac{1}{24}x^2 y^3\right)^2 \cdot (-64y)=- \frac{3\cdot64}{24} \cdot x^{5+4} \cdot y^{6+1} = -\frac{3\cdot8^2}{3\cdot8} x^9 y^7 = -8x^9 y^7 $
b) $ (-2ab)^3\cdot \underbrace{0,125}_{=1/8\text{}} a^2 c = -2^3\cdot \frac{1}{8} \cdot a^{3+2} b^3 c = -a^5 b^3 c $
c) $ \left(1\frac{2}{3}bz^7\right)^5 \cdot \left(-\frac{3}{5}az\right)^4 = \left(\frac{5}{3}\right)^5 \cdot \left(-\frac{3}{5}\right)^4 \cdot a^4 b^5 z^{5\cdot7+4} = \frac{5}{3} a^4 b^5 z^{39} $
d) $ (-0,5m^2 n^5 )^2 \cdot 12mn^3 = \left(-\frac{1}{2}\right)^2 \cdot 12 \cdot m^{4+1} \cdot n^{10+3} = 3m^5 n^{13} $
Example 4*. Find the value of n that make the equality true:
a) $ \left(\frac{1}{6}xy\right)^n \cdot 72x = 2x^3 y^2 $
Write the equations for the coefficients and powers on the left and on the right:
$$ {\left\{ \begin{array}{c} \left(\frac{1}{6}\right)^n \cdot 72 = 2 \\ x^{n+1} = x^3 \\ y^n = y^2 \end{array} \right.} $$
Thus, n = 2. Check:
$$ \left(\frac{1}{6}xy\right)^2 \cdot 72x = \left(\frac{1}{6}\right)^2 \cdot 72 \cdot x^{2+1} \cdot y^2 = 2x^3 y^2 $$
Answer: n=2
b) $ (-1 \frac{1}{3} a^2 b)^n \cdot 0,75b^3 = 1 \frac{1}{3} a^4 b^5 $
Write the equations for the coefficients and powers on the left and on the right:
$$ {\left\{ \begin{array}{c} \left(-1\frac{1}{3}\right)^n \cdot 0,75 = 1\frac{1}{3} \\ a^{2n} = a^4 \\ b^{n+3} = b^5 \end{array} \right.} $$
Thus, n = 2. Check:
$$ \left(-1\frac{1}{3}a^2b\right)^2 \cdot 0,75b^3 = \left(-\frac{4}{3}\right)^2 \cdot \left(\frac{3}{4}\right) \cdot a^4 \cdot b^{2+3} = \frac{4}{3}a^4b^5 = 1\frac{1}{3}a^4b^5 $$
Answer: n = 2
