Simplify each expression. (6m^2 n^3 )^2/(-4m^7 n^5 )^3 ( 6 m 2 n 3 ) 2 ( − 4 m 7 n 5 ) 3 \frac{(6m^2n^3)^2}{(-4m^7n^5)^3}

This problem, we want to simplify the expression six times m squared times n cubed all to the second power divided by negative four times m to the seventh power times n to the fifth power all cubed. We're going to use a couple of rules in order to do this. So first, let's use our power of a product rule to distribute in these exponents to each of the expressions on the top and bottom of our fraction. So for the top, we'd end up with six squared times m squared squared times n cubed squared. And on the bottom, we'll distribute in the three to each of our factors. And we'll get negative one cubed times four cubed times m to the seventh cubed times n to the fifth cubed. And now we're going to use our power rule to simplify some of these factors. So first, let's evaluate six squared as 36. M squared squared becomes m to the fourth, n cubed squared becomes n to the sixth, and that will be divided by negative one to the third power is negative one, since three is odd. Four cubed is 64, m to the seventh cubed is m to the twenty first power, and n to the fifth power cubed is n to the fifteenth power. Now we can continue simplifying by simplifying our constants first. So I know that both thirty six and sixty four are divisible by four. If I divide 36 by four, I'll get nine. And if I divide 64 by four, I'll get 16. So my constants will simplify to nine over negative 16. Now, looking at my variables, I can use my quotient rule. We know from the quotient rule that if we have two variables of the same base being divided, we subtract the exponents. So, m to the fourth over m to the twenty first becomes m to the negative seventeenth, and n to the sixth divided by n to the becomes n to the negative ninth. And this is now our answer in its simplest form.

Simplify each expression. (6m^2 n^3 )^2/(-4m^7 n^5 )^3 ( 6 m 2 n 3 ) 2 ( − 4 m 7 n 5 ) 3 \frac{(6m^2n^3)^2}{(-4m^7n^5)^3}

− 3 2 m − 5 n 2 -\frac32m^{-5}n^2

6. Exponents, Polynomials, and Polynomial Functions

Intro to the Power Rules

Intermediate Algebra

6. Exponents, Polynomials, and Polynomial Functions

Intro to the Power Rules

Struggling with Intermediate Algebra?

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Multiple Choice

Simplify each expression.
(6m^2 n^3 )^2/(-4m^7 n^5 )^3
$\frac{(6 m^{2} n^{3})^{2}}{(- 4 m^{7} n^{5})^{3}}$

$- \frac{9}{16} m^{17} n^{9}$

$- \frac{9}{16} m^{- 17} n^{- 9}$

$- \frac{9}{16} m^{- 17} n^{9}$

$- \frac{3}{2} m^{- 5} n^{2}$

Verified step by step guidance

Start by applying the power of a product rule to both the numerator and the denominator. This means raising each factor inside the parentheses to the given exponent: write $(6m^2n^3)^2$ as \$6^2 \cdot (m^2)^2 \cdot (n^3)^2$ and $(-4m^7n^5)^3$ as $(-4)^3 \cdot (m^7)^3 \cdot (n^5)^3$.

Simplify each part by calculating the powers: \$6^2$, $(-4)^3$, and use the power of a power rule for the variables, which states $(a^m)^n = a^{m \cdot n}$. So, $(m^2)^2 = m^{2 \cdot 2}$, $(n^3)^2 = n^{3 \cdot 2}$, $(m^7)^3 = m^{7 \cdot 3}$, and $(n^5)^3 = n^{5 \cdot 3}$.

Rewrite the expression with these simplified powers: the numerator becomes \$6^2 m^{4} n^{6}$ and the denominator becomes $(-4)^3 m^{21} n^{15}$.

Next, write the entire expression as a fraction: $\frac{6^2 m^{4} n^{6}}{(-4)^3 m^{21} n^{15}}$. Then, separate the constants and the variables: $\frac{6^2}{(-4)^3} \cdot \frac{m^{4}}{m^{21}} \cdot \frac{n^{6}}{n^{15}}$.

Finally, apply the quotient rule for exponents, which states $\frac{a^m}{a^n} = a^{m-n}$, to simplify the variables: $m^{4-21} = m^{-17}$ and $n^{6-15} = n^{-9}$. Combine the constants and variables to write the simplified expression.

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Simplify each expression.(6m^2 n^3 )^2/(-4m^7 n^5 )^3 (6m2n3)2(−4m7n5)3\frac{(6m^2n^3)^2}{(-4m^7n^5)^3}

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Simplify each expression.
(6m^2 n^3 )^2/(-4m^7 n^5 )^3
$\frac{(6 m^{2} n^{3})^{2}}{(- 4 m^{7} n^{5})^{3}}$