Sure, let's address the problem step-by-step and multiply the given binomials, then simplify the result.
Given binomials:
[tex]\[ \left(\left(x^2 8^2 c \cdot q^8 \cdot 15 \cdot x^2 \cdot c^3 \cdot c^3\right)\left(2 x \cdot x^8 \cdot 1^8 \cdot 5 \cdot c^8\right)\right)^8 \][/tex]
First, let's simplify the individual binomials within the parentheses.
### Simplifying the first binomial:
[tex]\[ x^2 \cdot 8^2 \cdot c \cdot q^8 \cdot 15 \cdot x^2 \cdot c^3 \cdot c^3 \][/tex]
Combine the like terms:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ x^2 \cdot x^2 = x^4 \][/tex]
[tex]\[ c \cdot c^3 \cdot c^3 = c^7 \][/tex]
[tex]\[ 15 \][/tex]
Therefore:
[tex]\[ 64 \cdot 15 \cdot x^4 \cdot c^7 \cdot q^8 \][/tex]
Calculate the constant term:
[tex]\[ 64 \cdot 15 = 960 \][/tex]
So the simplified first binomial is:
[tex]\[ 960 \cdot x^4 \cdot c^7 \cdot q^8 \][/tex]
### Simplifying the second binomial:
[tex]\[ 2 x \cdot x^8 \cdot 1^8 \cdot 5 \cdot c^8 \][/tex]
Combine the like terms:
[tex]\[ x \cdot x^8 = x^9 \][/tex]
[tex]\[ 1^8 = 1 \][/tex]
[tex]\[ 2 \cdot 5 = 10 \][/tex]
[tex]\[ c^8 \][/tex]
Therefore:
[tex]\[ 10 \cdot x^9 \cdot c^8 \][/tex]
### Multiplying the simplified binomials:
Now multiply the simplified binomials:
[tex]\[ \left(960 \cdot x^4 \cdot c^7 \cdot q^8 \cdot 10 \cdot x^9 \cdot c^8 \right)^8 \][/tex]
First combine like terms:
[tex]\[ 960 \cdot 10 = 9600 \][/tex]
[tex]\[ x^4 \cdot x^9 = x^{13} \][/tex]
[tex]\[ c^7 \cdot c^8 = c^{15} \][/tex]
Thus:
[tex]\[ (9600 \cdot x^{13} \cdot c^{15} \cdot q^8)^8 \][/tex]
### Raising to the 8th power:
When you raise a product to a power, you raise each factor to that power:
[tex]\[ 9600^8 \cdot (x^{13})^8 \cdot (c^{15})^8 \cdot (q^8)^8 \][/tex]
Now simplify the exponents:
[tex]\[ x^{13 \cdot 8} = x^{104} \][/tex]
[tex]\[ c^{15 \cdot 8} = c^{120} \][/tex]
[tex]\[ q^{8 \cdot 8} = q^{64} \][/tex]
Finally:
[tex]\[ 9600^8 \cdot x^{104} \cdot c^{120} \cdot q^{64} \][/tex]
Therefore, the multiplied and simplified form of the given binomials is:
[tex]\[ 9600^8 \cdot x^{104} \cdot c^{120} \cdot q^{64} \][/tex]
