Let's simplify each of the expressions one by one:
a) ( x^5 \cdot x^6 \cdot x^2 )
Using the property of exponents that states ( a^m \cdot a^n = a^{m+n} ):
[
x^5 \cdot x^6 \cdot x^2 = x^{5 + 6 + 2} = x^{13}
]
b) ( a^{12} : a^{10} : a )
First, we interpret the colon ":" as a sequence of divisions. This can be represented as:
[
\frac{a^{12}}{a^{10}} \cdot \frac{1}{a}
]
Using the property of exponents for division, ( \frac{a^m}{a^n} = a^{m-n} ):
[
\frac{a^{12}}{a^{10}} = a^{12-10} = a^2
]
Now, the next part:
[
\frac{a^2}{a} = a^{2-1} = a^1 = a
]
c) ( (2x^3)^2 )
Using the power of a product property ( (ab)^n = a^n \cdot b^n ):
[
(2x^3)^2 = 2^2 \cdot (x^3)^2 = 4 \cdot x^{3 \cdot 2} = 4x^6
]
d) ( (-3x^3y^2)^4 )
Again, using the power of a product property:
[
(-3x^3y^2)^4 = (-3)^4 \cdot (x^3)^4 \cdot (y^2)^4
]
Calculating each part:
[
(-3)^4 = 81, \quad (x^3)^4 = x^{3 \cdot 4} = x^{12}, \quad (y^2)^4 = y^{2 \cdot 4} = y^8
]
Thus, combining these results:
[
(-3x^3y^2)^4 = 81x^{12}y^8
]
To summarize:
a) ( x^{13} )
b) ( a )
c) ( 4x^6 )
d) ( 81x^{12}y^8 )
