2. Simplity each of the following, expressing your answer in positive index form. $$ \begin{array}{ll} ext { (a) } a^{5} \div a^{-2} & ext { (b) } b^{4} \div \sqrt{b} imes b^{-7} \ ext { (c) }\left(m^{2} n ight)^{4} imes\left(m^{-4} n^{3} ight)^{5} & ext { (d) }\left(\frac{x^{-3} y}{x^{2} y^{-2}} ight)^{-5}\end{array} $$

Question

2. Simplity each of the following, expressing your answer in positive index form. $$ \begin{array}{ll}	ext { (a) } a^{5} \div a^{-2} & 	ext { (b) } b^{4} \div \sqrt{b} 	imes b^{-7} \ 	ext { (c) }\left(m^{2} night)^{4} 	imes\left(m^{-4} n^{3}ight)^{5} & 	ext { (d) }\left(\frac{x^{-3} y}{x^{2} y^{-2}}ight)^{-5}\end{array} $$

UpStudy AI · Accepted Answer

**(a)**  
$$
a^5 \div a^{-2} = a^{5 - (-2)} = a^{5+2} = a^7.
$$

**(b)**  
Write $$\sqrt{b}$$ as $$b^{1/2}$$:
$$
b^4 \div b^{1/2} \times b^{-7} = b^{4 - \frac{1}{2} - 7} = b^{\frac{8}{2} - \frac{1}{2} - \frac{14}{2}} = b^{\frac{8 - 1 - 14}{2}} = b^{-\frac{7}{2}}.
$$
Expressing with positive indices:
$$
b^{-\frac{7}{2}} = \frac{1}{b^{\frac{7}{2}}}.
$$

**(c)**  
Raise each term to the indicated power:
$$
\left(m^2n\right)^4 = m^{2 \times 4}n^4 = m^8 n^4,
$$
$$
\left(m^{-4}n^3\right)^5 = m^{-4 \times 5}n^{3 \times 5} = m^{-20} n^{15}.
$$
Multiply the results:
$$
m^8 n^4 \times m^{-20} n^{15} = m^{8 + (-20)} n^{4+15} = m^{-12} n^{19}.
$$
Expressing with positive index form:
$$
m^{-12} n^{19} = \frac{n^{19}}{m^{12}}.
$$

**(d)**  
First, simplify the expression inside the parentheses:
$$
\frac{x^{-3}y}{x^2y^{-2}} = x^{-3-2}y^{1-(-2)} = x^{-5}y^{3}.
$$
Now, raise this to the power $$-5$$:
$$
\left(x^{-5}y^3\right)^{-5} = x^{-5 \times (-5)}y^{3 \times (-5)} = x^{25}y^{-15}.
$$
Expressing with positive indices:
$$
x^{25}y^{-15} = \frac{x^{25}}{y^{15}}.
$$
**(a)** $$ a^7 $$ **(b)** $$ \frac{1}{b^{\frac{7}{2}}} $$ **(c)** $$ \frac{n^{19}}{m^{12}} $$ **(d)** $$ \frac{x^{25}}{y^{15}} $$

Simplity each of the following, expressing your
answer in positive index form.

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