Math Snap

STEP 1

Assumptions1. All variables are non-zero.
. The expressions need to be simplified and expressed with positive indices.

STEP 2

Let's start with the first expression. We'll use the power of a power rule, which states that $(a^m)^n = a^{mn}$.
\left( w^{6}\right)^{} =^ \cdot (w^6)^

STEP 3

Calculate the power of3 and apply the power of a power rule to $w^6$.
$$\left(3 w^{6}\right)^{3} =27 \cdot w^{6 \cdot3}$$

STEP 4

implify the expression.
$$\left(3 w^{6}\right)^{3} =27w^{18}$$

STEP 5

Now, let's move to the second expression. Again, we'll use the power of a power rule.
$$\left(2^{2} s^{5}\right)^{3} = (2^2)^3 \cdot (s^5)^3$$

STEP 6

Calculate the power of2 and apply the power of a power rule to $s^5$.
$$\left(2^{2} s^{5}\right)^{3} =2^{2 \cdot3} \cdot s^{5 \cdot3}$$

STEP 7

implify the expression.
$$\left(2^{2} s^{5}\right)^{3} =64s^{15}$$

STEP 8

For the third expression, we'll use the power of a fraction rule, which states that $(\frac{a}{b})^n = \frac{a^n}{b^n}$.
$$\left(\frac{p}{q^{5}}\right)^{3} = \frac{p^3}{(q^5)^3}$$

STEP 9

Apply the power of a power rule to $q^5$.
$$\left(\frac{p}{q^{5}}\right)^{3} = \frac{p^3}{q^{5 \cdot3}}$$

STEP 10

implify the expression.
$$\left(\frac{p}{q^{5}}\right)^{3} = \frac{p^3}{q^{15}}$$

STEP 11

For the fourth expression, we'll again use the power of a fraction rule.
$$\left(\frac{-m^{3}}{n^{5}}\right)^{5} = \frac{(-m^3)^5}{(n^5)^5}$$

STEP 12

Apply the power of a power rule to -m^ and $n^5$.
$$\left(\frac{-m^{}}{n^{5}}\right)^{5} = \frac{-m^{ \cdot5}}{n^{5 \cdot5}}$$

STEP 13

implify the expression.
$$\left(\frac{-m^{3}}{n^{5}}\right)^{5} = -\frac{m^{15}}{n^{25}}$$

STEP 14

For the fifth expression, we'll use the rule that $a^{-n} = \frac{}{a^n}$.
\left(s^{3}\right)^{-} \times s^{7} = \frac{}{(s^3)^} \times s^7

STEP 15

Apply the power of a power rule to $s^3$.
$$\left(s^{3}\right)^{-5} \times s^{7} = \frac{}{s^{3 \cdot5}} \times s^7$$

STEP 16

implify the expression.
\left(s^{3}\right)^{-5} \times s^{} = \frac{s^}{s^{15}}

STEP 17

Use the rule that $\frac{a^m}{a^n} = a^{m-n}$.
$$\left(s^{3}\right)^{-5} \times s^{7} = s^{7-15}$$

STEP 18

implify the expression.
$$\left(s^{3}\right)^{-5} \times s^{7} = s^{-8}$$

STEP 19

For the sixth expression, we'll use the rule that $a^{-n} = \frac{1}{a^n}$ and $\frac{a^m}{a^n} = a^{m-n}$.
u^{-4} \div\left(u^{5}\right)^{-} = \frac{1}{u^4} \div \frac{1}{(u^5)^}

STEP 20

Apply the power of a power rule to $u^5$.
$$u^{-4} \div\left(u^{5}\right)^{-} = \frac{}{u^4} \div \frac{}{u^{5 \cdot}}$$

STEP 21

implify the expression.
$$u^{-4} \div\left(u^{5}\right)^{-} = \frac{1}{u^4} \div \frac{1}{u^{10}}$$

STEP 22

Use the rule that $\frac{a}{\frac{b}{c}} = a \cdot c$.
$$u^{-4} \div\left(u^{5}\right)^{-} = \frac{1}{u^4} \cdot u^{10}$$

STEP 23

Use the rule that $\frac{a^m}{a^n} = a^{m-n}$.
$$u^{-} \div\left(u^{5}\right)^{-} = u^{10-}$$

STEP 24

implify the expression.
$$u^{-4} \div\left(u^{}\right)^{-} = u^{6}$$

STEP 25

For the seventh expression, we'll use the rule that $a^{-n} = \frac{1}{a^n}$ and $\frac{a^m}{a^n} = a^{m-n}$.
\frac{d^{3} \div\left(d^{-5}\right)^{}}{d^{-}} = \frac{d^3 \div \frac{1}{(d^{-5})^}}{\frac{1}{d^}}

STEP 26

Apply the power of a power rule to $d^{-5}$.
\frac{d^{3} \div\left(d^{-5}\right)^{}}{d^{-}} = \frac{d^3 \div \frac{1}{d^{-5 \cdot}}}{\frac{1}{d^}}

STEP 27

implify the expression.
\frac{d^{3} \div\left(d^{-5}\right)^{}}{d^{-}} = \frac{d^3 \div \frac{1}{d^{-10}}}{\frac{1}{d^}}

STEP 28

Use the rule that $\frac{a}{\frac{b}{c}} = a \cdot c$.
\frac{d^{3} \div\left(d^{-5}\right)^{}}{d^{-}} = d^3 \cdot d^{10} \cdot d^

STEP 29

Use the rule that $a^m \cdot a^n = a^{m+n}$.
$$\frac{d^{} \div\left(d^{-5}\right)^{2}}{d^{-2}} = d^{+10+2}$$

STEP 30

implify the expression.
$$\frac{d^{} \div\left(d^{-5}\right)^{2}}{d^{-2}} = d^{15}$$

STEP 31

For the eighth expression, we'll use the rule that $a^{-n} = \frac{1}{a^n}$.
\frac{1}{(x y)^{-}} = \frac{1}{\frac{1}{(x y)^}}

STEP 32

Use the rule that $\frac{a}{\frac{b}{c}} = a \cdot c$.
\frac{1}{(x y)^{-}} =1 \cdot (x y)^

implify the expression.
\frac{1}{(x y)^{-}} = (x y)^The simplified expressions are11. $27w^{18}$
12. $64s^{15}$
13. \frac{p^}{q^{15}}
14. $-\frac{m^{15}}{n^{25}}$
15. $s^{-8}$
16. $u^{6}$
17. $d^{15}$
18. (x y)^