Here are the solutions to the problems using exponent rules. 2. Use the exponent rules to simplify in exponential form. a)* $8^4 + 8^2$ This expression cannot be simplified into a single exponential term using exponent rules because it is an addition of terms with different exponents. b)* $7^{12} \div 7^4$ Step 1: Apply the division rule for exponents, $a^m \div a^n = a^{m-n}$. $$7^{12-4}$$ Step 2: Simplify the exponent. $$7^8$$ The simplified form is $\boxed{7^8}$. c)* $b^5 + b^3$ This expression cannot be simplified into a single exponential term using exponent rules because it is an addition of terms with different exponents. d)* $9^8 / 9^6$ Step 1: Apply the division rule for exponents, $a^m / a^n = a^{m-n}$. $$9^{8-6}$$ Step 2: Simplify the exponent. $$9^2$$ The simplified form is $\boxed{9^2}$. e)* $12^7 / 12^3$ Step 1: Apply the division rule for exponents, $a^m / a^n = a^{m-n}$. $$12^{7-3}$$ Step 2: Simplify the exponent. $$12^4$$ The simplified form is $\boxed{12^4}$. f)* $y^8 / y^4$ Step 1: Apply the division rule for exponents, $a^m / a^n = a^{m-n}$. $$y^{8-4}$$ Step 2: Simplify the exponent. $$y^4$$ The simplified form is $\boxed{y^4}$. 3. Use the exponent rules to simplify these, and leave your answer in exponential form. a)* $(8^6)^2$ Step 1: Apply the power of a power rule, $(a^m)^n = a^{m \times n}$. $$8^{6 \times 2}$$ Step 2: Simplify the exponent. $$8^{12}$$ The simplified form is $\boxed{8^{12}}$. b)* $(9^{12})^4$ Step 1: Apply the power of a power rule, $(a^m)^n = a^{m \times n}$. $$9^{12 \times 4}$$ Step 2: Simplify the exponent. $$9^{48}$$ The simplified form is $\boxed{9^{48}}$. c)* $(a^5)^3$ Step 1: Apply the power of a power rule, $(a^m)^n = a^{m \times n}$. $$a^{5 \times 3}$$ Step 2: Simplify the exponent. $$a^{15}$$ The simplified form is $\boxed{a^{15}}$. d)* $(p^2)^2$ Step 1: Apply the power of a power rule, $(a^m)^n = a^{m \times n}$. $$p^{2 \times 2}$$ Step 2: Simplify the exponent. $$p^4$$ The simplified form is $\boxed{p^4}$. e)* $2(a^{10})^3$ Step 1: Apply the power of a power rule to $(a^{10})^3$. $$2 \times a^{10 \times 3}$$ Step 2: Simplify the exponent. $$2a^{30}$$ The simplified form is $\boxed{2a^{30}}$. f)* $3^2(y^4)^3$ Step 1: Apply the power of a power rule to $(y^4)^3$. $$3^2 \times y^{4 \times 3}$$ Step 2: Simplify the exponent. $$3^2 y^{12}$$ Step 3: Calculate $3^2$. $$9y^{12}$$ The simplified form is $\boxed{9y^{12}}$. 4. Determine these values. a)* $(-4)^3 + (3^2)^3$ Step 1: Calculate $(-4)^3$. $$(-4)^3 = -4 \times -4 \times -4 = -64$$ Step 2: Calculate $(3^2)^3$. $$(3^2)^3 = (9)^3 = 9 \times 9 \times 9 = 729$$ Step 3: Add the results. $$-64 + 729 = 665$$ The value is $\boxed{665}$. b)* $3^4 \div 3^3 + \sqrt{8}$ Step 1: Simplify $3^4 \div 3^3$ using the division rule for exponents. $$3^{4-3} = 3^1 = 3$$ Step 2: Simplify $\sqrt{8}$. $$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$$ Step 3: Add the results. $$3 + 2\sqrt{2}$$ The value is $\boxed{3 + 2\sqrt{2}}$. c)* $4^2 + 4^1 + 4^0$ Step 1: Calculate each term. $$4^2 = 16$$ $$4^1 = 4$$ $$4^0 = 1$$ Step 2: Add the results. $$16 + 4 + 1 = 21$$ The value is $\boxed{21}$. d)* $3^{10} \div 3^5 \div 3^2$ Step 1: Apply the division rule for exponents from left to right. $$3^{10-5-2}$$ Step 2: Simplify the exponent. $$3^{5-2} = 3^3$$ Step 3: Calculate $3^3$. $$3^3 = 3 \times 3 \times 3 = 27$$ The value is $\boxed{27}$. e)* $10^2 \times 10^1 \times 10^0 - 10^3$ Step 1: Simplify $10^2 \times 10^1 \times 10^0$ using the multiplication rule for exponents, $a^m \times a^n = a^{m+n}$. $$10^{2+1+0} = 10^3$$ Step 2: Subtract $10^3$. $$10^3 - 10^3 = 0$$ The value is $\boxed{0}$. f)* $3a^2 \