Expressions and Equations: Using Properties of Integer Exponents (CCSS.8.EE.1)

Use the product of powers: add exponents with the same base. $2 + (-5) = -3$, so $3^2 \times 3^{-5} = 3^{-3}$. $3^{7}$ incorrectly adds $2+5$. $3^{3}$ drops the negative sign or subtracts in the wrong order. $3^{-10}$ multiplies exponents instead of adding.

Use the quotient of powers: subtract exponents with the same base. $6-2=4$, so $\frac{5^6}{5^2}=5^4$. $5^{12}$ multiplies exponents. $5^{8}$ adds exponents. $5^{-4}$ subtracts in the wrong order ($2-6$).

First apply power of a power: $(x^3)^2 = x^{3\cdot 2}=x^6$. Then use the product rule: $x^6 \cdot x^{-4} = x^{6+(-4)}=x^2$. $x$ treats $(x^3)^2$ as $x^{3+2}=x^5$ then $5+(-4)=1$. $x^{5}$ ignores the $x^{-4}$ factor. $x^{-2}$ subtracts in the wrong order ($-4-6$).

Subtract exponents for each base: $x^{5-(-1)}=x^{6}$ and $y^{-2-3}=y^{-5}$, so the result is $x^6 y^{-5}=\frac{x^6}{y^5}$. $x^{4} y^{-5}$ subtracts $x$ exponents incorrectly ($5-1$ instead of $5-(-1)$). $x^{4} y^{5}$ also drops the negative on $y$. $x^{-6} y^{5}$ flips signs as if swapping numerator and denominator.

Use the product of powers rule: $a^m\cdot a^n=a^{m+n}$. So $2^3\times 2^4=2^{3+4}=2^7$. A multiplies exponents ($3\times 4$) instead of adding. C subtracts exponents as if dividing ($3-4$). D ignores the $2^4$ factor.

Use the quotient of powers rule: $\dfrac{a^m}{a^n}=a^{m-n}$. So $\dfrac{3^5}{3^2}=3^{5-2}=3^3$. A multiplies exponents ($5\times 2$). B adds exponents instead of subtracting. C subtracts in the wrong order ($2-5$), giving a negative exponent.

Use the power of a power rule: $(a^m)^n=a^{mn}$. Thus $(x^3)^2=x^{3\cdot 2}=x^6$. B adds exponents ($3+2$). C ignores the outer exponent. D treats it as $x^{3^2}=x^9$, which is not the rule for a power of a power.

Use the product of powers rule: add exponents. $-3+2=-1$, so $10^{-3}\times 10^{2}=10^{-1}$. A subtracts exponents as if dividing. B ignores the negative sign and adds absolute values. D multiplies the exponents ($-3\times 2$) instead of adding.

Use the product rule: $a^m\cdot a^n=a^{m+n}$. Here, $2^4\cdot 2^{-7}=2^{4+(-7)}=2^{-3}$. Choice B adds magnitudes or subtracts in the wrong way to get $11$. Choice C drops the negative sign. Choice D multiplies the exponents ($4\times -7$), which is not a valid rule.

Use the quotient rule: $\dfrac{a^m}{a^n}=a^{m-n}$. So $\dfrac{x^5}{x^2}=x^{5-2}=x^3$. Choice A multiplies exponents. Choice C adds exponents instead of subtracting. Choice D subtracts in the wrong order ($2-5$).