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Hai can anyone help

Hai can anyone help-example-1
User Bkulyk
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2 Answers

2 votes

Answer:


\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)}}

User Ahron
by
8.5k points
4 votes

Answer:


\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.

User Mantas
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