Here are the solutions to the questions: 1. Simplify the expression $\frac{30x^6}{3x^{-2}}$ Working out: Step 1: Divide the numerical coefficients. $$ \frac{30}{3} = 10 $$ Step 2: Apply the division rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. $$ \frac{x^6}{x^{-2}} = x^{6 - (-2)} = x^{6+2} = x^8 $$ Step 3: Combine the simplified numerical and variable terms. $$ 10x^8 $$ Answer: $$ \boxed{10x^8} $$ 2. Simplify $\left(\frac{x^3}{2^2}\right)^{-2}$ Working out: Step 1: Apply the negative exponent rule, which states that $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$. $$ \left(\frac{x^3}{2^2}\right)^{-2} = \left(\frac{2^2}{x^3}\right)^2 $$ Step 2: Apply the power of a quotient rule, which states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. $$ \left(\frac{2^2}{x^3}\right)^2 = \frac{(2^2)^2}{(x^3)^2} $$ Step 3: Apply the power of a power rule, which states that $(a^m)^n = a^{mn}$. $$ \frac{(2^2)^2}{(x^3)^2} = \frac{2^{2 \times 2}}{x^{3 \times 2}} = \frac{2^4}{x^6} $$ Step 4: Calculate the numerical value $2^4$. $$ 2^4 = 2 \times 2 \times 2 \times 2 = 16 $$ Step 5: Substitute the numerical value back into the expression. $$ \frac{16}{x^6} $$ Answer: $$ \boxed{\frac{16}{x^6}} $$ 3. Simplify the powers: $\frac{x^6y^8z^{-3}}{x^{-1}z^{-5}}$ Working out: Step 1: Apply the division rule for exponents ($\frac{a^m}{a^n} = a^{m-n}$) for each variable. For $x$: $$ \frac{x^6}{x^{-1}} = x^{6 - (-1)} = x^{6+1} = x^7 $$ For $y$: The $y^8$ term is only in the numerator, so it remains as $y^8$. For $z$: $$ \frac{z^{-3}}{z^{-5}} = z^{-3 - (-5)} = z^{-3+5} = z^2 $$ Step 2: Combine the simplified terms. $$ x^7y^8z^2 $$ Answer: $$ \boxed{x^7y^8z^2} $$ Last free one today — make it count tomorrow, or type /upgrade for unlimited.