Use Product of Powers Property: a) $(x^{3})(x^{(-2)/(3)})$ $\newline$ Use the product of powers property: $a^n \ast a^m = a^{(n+m)}$ $\newline$ $x^{(3 + (-2/3))} = x^{(9/3 - 2/3)} = x^{(7/3)}$
Use Power of a Power Property: b) $(81^{-0.25})^3$ $\newline$ Use the power of a power property: $(a^n)^m = a^{n*m}$ $\newline$ $81^{-0.25 * 3} = 81^{-0.75}$
Use Power of a Power Property: c) $((m^{-2})^{(2)/(3)})/((m^{(1)/(2)})^{4})$ $\newline$ Use the power of a power property: $(a^n)^m = a^{n*m}$ $\newline$ $m^{(-2)*(2/3)} / m^{(1/2)*4} = m^{-4/3} / m^2$ $\newline$ Use the quotient of powers property: $a^n / a^m = a^{n-m}$ $\newline$ $m^{-4/3 - 2} = m^{-4/3 - 6/3} = m^{-10/3}$
Use Power of a Product Property: d) $(9p^{2})^{-(1)/(2)})(p^{-(3)/(2)})$ $\newline$ Use the power of a product property: $(a*b)^{n} = a^{n} * b^{n}$ $\newline$ $(9^{-(1)/(2)})(p^{2*(-1)/(2)})(p^{-(3)/(2)})$ $\newline$ Simplify each term: $\newline$ $9^{(-1/2)} * p^{(-1)} * p^{(-3/2)}$ $\newline$ Combine the p terms using the product of powers property: $\newline$ $9^{(-1/2)} * p^{(-1 - 3/2)} = 9^{(-1/2)} * p^{(-5/2)}$
Use Power of a Quotient Property: e) $[(x^{-2})/((xy)^{4})]^{1.5}$ Use the power of a quotient property: $(a/b)^n = a^n / b^n$ $(x^{-2\cdot1.5})/((xy)^{4\cdot1.5})$ Simplify each term: $x^{-3} / (x^{6}y^{6})$ Use the quotient of powers property: $x^{-3 - 6} / y^{6} = x^{-9} / y^{6}$
Use Power of a Quotient Property: f) $[(4x^{-2})/(9y^{-4})]^{-\frac{5}{2}}$ $\newline$ Use the power of a quotient property: $(a/b)^n = a^n / b^n$ $\newline$ $(4^{-\frac{5}{2}})(x^{-2*(-\frac{5}{2})})/(9^{-\frac{5}{2}})(y^{-4*(-\frac{5}{2})})$ $\newline$ Simplify each term: $\newline$ $4^{-\frac{5}{2}} * x^{5} / 9^{-\frac{5}{2}} * y^{20}$ $\newline$ Combine the terms using the product of powers property: $\newline$ $(4^{-\frac{5}{2}} / 9^{-\frac{5}{2}}) * (x^{5} * y^{20})$
Use Power of a Quotient Property: a) $(x^{p})^{(1)/(3)}=x^{(2)/(3)}$ $\newline$ Use the power of a power property: $(a^n)^m = a^{n*m}$ $\newline$ $x^{p*(1/3)} = x^{2/3}$ $\newline$ Set the exponents equal to each other to find $p$ : $\newline$ $p*(1/3) = 2/3$ $\newline$ Multiply both sides by $3$ to solve for $p$ : $\newline$ $p = 2$
Use Power of a Product Property: b) $(x^{p})(x^{\frac{3}{4}})=x^{2}$ Use the product of powers property: $a^n \cdot a^m = a^{n+m}$ $x^{p + \frac{3}{4}} = x^{2}$ Set the exponents equal to each other to find $p$ : $p + \frac{3}{4} = 2$ Subtract $\frac{3}{4}$ from both sides to solve for $p$ : $p = 2 - \frac{3}{4} = 1 \frac{1}{4}$ or $\frac{5}{4}$
Use Power of a Quotient Property: c) $(x^{p})/(x^{-2})=x^{(5)/(2)}$ $\newline$ Use the quotient of powers property: $a^n / a^m = a^{(n-m)}$ $\newline$ $x^{(p - (-2))} = x^{(5/2)}$ $\newline$ Set the exponents equal to each other to find $p$ : $\newline$ $p + 2 = 5/2$ $\newline$ Subtract $2$ from both sides to solve for $p$ : $\newline$ $p = 5/2 - 4/2 = 1/2$
Use Property of Equality: d) $(-3x^{(5)/(2)})(px^{-(1)/(2)})=(-3)/(4)x^{2}$ $\newline$ Distribute the x terms: $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $\newline$ Simplify the x terms: $\newline$ $-3p \cdot x^{4/2} = -3/4 \cdot x^{2}$ $\newline$ $-3p \cdot x^{2} = -3/4 \cdot x^{2}$ $\newline$ Set the coefficients equal to each other to find p: $\newline$ $-3p = -3/4$ $\newline$ Divide both sides by $-3$ to solve for p: $\newline$ $p = 1/4$
Use Property of Equality: d) $(-3x^{(5)/(2)})(px^{-(1)/(2)})=(-3)/(4)x^{2}$ $\newline$ Distribute the x terms: $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $\newline$ Simplify the x terms: $\newline$ $-3p \cdot x^{4/2} = -3/4 \cdot x^{2}$ $\newline$ $-3p \cdot x^{2} = -3/4 \cdot x^{2}$ $\newline$ Set the coefficients equal to each other to find $p$ : $\newline$ $-3p = -3/4$ $\newline$ Divide both sides by $-3$ to solve for $p$ : $\newline$ $p = 1/4$ e) $((9a^{-4})/(25))^{p}=(3)/(5a^{2})$ $\newline$ Use the power of a quotient property: $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $0$ $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $1$ $\newline$ Set the $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $2$ terms equal to each other to find $p$ : $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $4$ $\newline$ Set the exponents equal to each other to find $p$ : $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $6$ $\newline$ Divide both sides by $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $7$ to solve for $p$ : $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $9$
Use Property of Equality: d) $(-3x^{(5)/(2)})(px^{-(1)/(2)})=(-3)/(4)x^{2}$ $\newline$ Distribute the x terms: $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $\newline$ Simplify the x terms: $\newline$ $-3p \cdot x^{4/2} = -3/4 \cdot x^{2}$ $\newline$ $-3p \cdot x^{2} = -3/4 \cdot x^{2}$ $\newline$ Set the coefficients equal to each other to find $p$ : $\newline$ $-3p = -3/4$ $\newline$ Divide both sides by $-3$ to solve for $p$ : $\newline$ $p = 1/4$ e) $((9a^{-4})/(25))^{p}=(3)/(5a^{2})$ $\newline$ Use the power of a quotient property: $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $0$ $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $1$ $\newline$ Set the $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $2$ terms equal to each other to find $p$ : $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $4$ $\newline$ Set the exponents equal to each other to find $p$ : $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $6$ $\newline$ Divide both sides by $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $7$ to solve for $p$ : $\newline$ $-3p \cdot x^{5/2 - 1/2} = -3/4 \cdot x^{2}$ $9$ f) $-3p \cdot x^{4/2} = -3/4 \cdot x^{2}$ $0$ $\newline$ Use the property of equality for exponents: $\newline$ $-3p \cdot x^{4/2} = -3/4 \cdot x^{2}$ $1$ $\newline$ Set the bases equal to each other to find $p$ : $\newline$ $-3p \cdot x^{4/2} = -3/4 \cdot x^{2}$ $3$ $\newline$ Since the bases are different, we cannot directly compare the exponents. This step is incorrect.