Question

Hai can anyone help

Answer 1

`\textsf{C)} \quad (12ac^(14))/(b^3)`

Explanation:

Given rational expression:

`(\left(a^2b^4c\right)^2\left(6a^3b\right)\left(2c^5\right)^3)/(4a^6b^(12)c^3)`

To simplify the given rational expression, we can use exponent rules.

`\textsf{Apply the exponent rule:} \quad \boxed{(x^p)^q=x^(pq)}`

`=(\left(a^((2\cdot 2))b^((4\cdot 2))c^((1\cdot 2))\right)\left(6a^3b\right)\left(2^((1\cdot 3))c^((5\cdot 3))\right))/(4a^6b^(12)c^3)`

`=(\left(a^(4)b^(8)c^(2)\right)\left(6a^3b\right)\left(2^(3)c^(15)\right))/(4a^6b^(12)c^3)`

`=(\left(a^(4)b^(8)c^(2)\right)\left(6a^3b\right)\left(8c^(15)\right))/(4a^6b^(12)c^3)`

Remove the brackets from the numerator:

`=(a^(4)b^(8)c^(2)\:6a^3b\:8c^(15))/(4a^6b^(12)c^3)`

Collect like terms in the numerator:

`=(6\cdot 8\:a^(4)a^3\:b^(8)b\:c^(2)c^(15))/(4a^6b^(12)c^3)`

Multiply the numbers:

`=(48\:a^(4)a^3\:b^(8)b\:c^(2)c^(15))/(4a^6b^(12)c^3)`

Divide the numbers:

`=(12\:a^(4)a^3\:b^(8)b\:c^(2)c^(15))/(a^6b^(12)c^3)`

`\textsf{Apply the exponent rule:} \quad \boxed{ x^p \cdot x^q=x^(p+q)}`

`=(12a^((4+3))b^((8+1))c^((2+15)))/(a^6b^(12)c^3)`

`=(12a^(7)b^(9)c^(17))/(a^6b^(12)c^3)`

`\textsf{Apply the exponent rule:} \quad \boxed{(x^p)/(x^q)=x^(p-q)}`

`=12a^((7-6))b^((9-12))c^((17-3))`

`=12a^((1))b^((-3))c^((14))`

`=12a\:b^(-3)c^(14)`

`\textsf{Apply the exponent rule:} \quad a^(-n)=(1)/(a^n)`

`=(12ac^(14))/(b^3)`

Therefore, the expression that is equivalent to the given rational expression is:

`\large\boxed{\boxed{(12ac^(14))/(b^3)}}`

Answer 2

`\tt c.\boxed{\tt (12ac^(14))/(b^3)}`

Explanation:

`\tt ((a^2b^4c)^2(6a^3b)(2c^5)^3 )/(4a^6b^(12)c^3)`

Distribute the exponents.

`\tt (a^4b^8c^2 * 6a^3b * 8c^(15))/(4a^6b^(12)c^3)`

Combining like terms.

`\tt (6*8*a^(4+3)b^(8+1)c^(2+15))/(4a^6b^(12)c^3)\\\\\tt (48*a^(7)b^(9)c^(17))/(4a^6b^(12)c^3)`

Cancel out common factors.

`\tt (12*a^(7-6)c^(17-3))/(b^(12-9))\\\\\tt (12ac^(14))/(b^3)`

Therefore, the simplified expression is
`\tt c.\boxed{\tt (12ac^(14))/(b^3)}`

Note:

For rule of indices:

There are the 7 laws of indices:

1. Power of a power:
`\boxed{\tt(a^m)^n = a^(mn). }`

2. Product of powers:
`\boxed{\tt a^m \cdot a^n = a^(m+n)}`

3. Quotient of powers:
`\boxed{\tt (a^m)/(a^n) = a^(m-n)}`

4. Power of a product:
`\boxed{\tt(ab)^m = a^m b^m}`

5. Power of a quotient:
`\boxed{\tt (a)/(b))^m = (a^m)/(b^m)}`

6. Zeroth power:
`\boxed{\tt a^0 = 1}`, for any non-zero real number a.

7. Negative power:
`\boxed{\tt a^(-m) = (1)/(a^m)}`, for any non-zero real number a and positive integer m.