Question

Возвести в степень (m^6/n^3)^2 (-3a/2b^2)^4 (-5a^3b^4/3c^5d^7)^3

Answer 1

$( \frac{m^{6} }{n^{3}})^{2} = \frac{ {m}^{12} }{n^{6} }$
$(\frac{ - 3a}{2 {b}^{2} } )^{4} = \frac{( - 3) ^{4}a^{4} }{ {2}^{4} {b}^{8} } = \frac{81 {a}^{4} }{16 {b}^{8} }$
$(\frac{ - 5a ^{3} 3 {b}^{4} }{3 {c}^{5} 5 {d}^{7} } )^{3} = - (\frac{15a ^{3} {b}^{4}}{ 15{c}^{5} {d}^{7} } ) ^{3} = - \frac{ {a}^{9} {b}^{12} }{ {c}^{15} {d}^{21} }$

Answer 2

To simplify the expression, let's take each term step by step:

1. $(m^6/n^3)^2$: When raising a power to another power, you multiply the exponents. So, $(m^6/n^3)^2 = m^{6 \times 2} / n^{3 \times 2} = m^{12} / n^6$.

2. $(-3a/2b^2)^4$: When raising a fraction to a power, you raise both the numerator and denominator to that power. So, $(-3a/2b^2)^4 = (-3^4 \cdot a^4) / (2^4 \cdot b^{2 \times 4}) = (81a^4) / (16b^8)$.

3. $(-5a^3b^4/3c^5d^7)^3$: When raising a fraction to a power, you raise both the numerator and denominator to that power. So, $(-5a^3b^4/3c^5d^7)^3 = (-5^3 \cdot a^{3 \times 3} \cdot b^{4 \times 3}) / (3^3 \cdot c^{5 \times 3} \cdot d^{7 \times 3}) = (-125a^9b^{12}) / (27c^{15}d^{21})$.

Now, let's combine all the simplified terms:

$(m^6/n^3)^2 \cdot (-3a/2b^2)^4 \cdot (-5a^3b^4/3c^5d^7)^3 = (m^{12} / n^6) \cdot (81a^4 / 16b^8) \cdot (-125a^9b^{12} / 27c^{15}d^{21})$.

To multiply the terms, we multiply the numerators and denominators separately:

Numerator: $m^{12} \cdot 81a^4 \cdot (-125a^9b^{12}) = -10125m^{12}a^{13}b^{12}$. Denominator: $n^6 \cdot 16b^8 \cdot 27c^{15}d^{21} = 432b^8c^{15}d^{21}$.

Now, the expression becomes:

$(-10125m^{12}a^{13}b^{12}) / (432b^8c^{15}d^{21})$.

Finally, we can simplify it further by canceling out common terms:

$-10125m^{12}a^{13}b^{12} / (432b^8c^{15}d^{21}) = - \frac{10125m^{12}a^{13}}{432b^{8}c^{15}d^{21}}$.

So, the simplified expression is $- \frac{10125m^{12}a^{13}}{432b^{8}c^{15}d^{21}}$.