Use the exponent rules to simplify in exponential form.

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 ÷ 7^4 Step 1: Apply the division rule for exponents, a^m ÷ a^n = a^m-n. 7^12-4 Step 2: Simplify the exponent. 7^8 The simplified form is 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 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 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 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 × n. 8^6 × 2 Step 2: Simplify the exponent. 8^12 The simplified form is 8^12. b)* (9^12)^4 Step 1: Apply the power of a power rule, (a^m)^n = a^m × n. 9^12 × 4 Step 2: Simplify the exponent. 9^48 The simplified form is 9^48. c)* (a^5)^3 Step 1: Apply the power of a power rule, (a^m)^n = a^m × n. a^5 × 3 Step 2: Simplify the exponent. a^15 The simplified form is a^15. d)* (p^2)^2 Step 1: Apply the power of a power rule, (a^m)^n = a^m × n. p^2 × 2 Step 2: Simplify the exponent. p^4 The simplified form is p^4. e)* 2(a^10)^3 Step 1: Apply the power of a power rule to (a^10)^3. 2 × a^10 × 3 Step 2: Simplify the exponent. 2a^30 The simplified form is 2a^30. f)* 3^2(y^4)^3 Step 1: Apply the power of a power rule to (y^4)^3. 3^2 × y^4 × 3 Step 2: Simplify the exponent. 3^2 y^12 Step 3: Calculate 3^2. 9y^12 The simplified form is 9y^12. 4. Determine these values. a)* (-4)^3 + (3^2)^3 Step 1: Calculate (-4)^3. (-4)^3 = -4 × -4 × -4 = -64 Step 2: Calculate (3^2)^3. (3^2)^3 = (9)^3 = 9 × 9 × 9 = 729 Step 3: Add the results. -64 + 729 = 665 The value is 665. b)* 3^4 ÷ 3^3 + sqrt(8) Step 1: Simplify 3^4 ÷ 3^3 using the division rule for exponents. 3^4-3 = 3^1 = 3 Step 2: Simplify sqrt(8). sqrt(8) = sqrt(4 × 2) = sqrt(4) × sqrt(2) = 2sqrt(2) Step 3: Add the results. 3 + 2sqrt(2) The value is 3 + 2sqrt(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 21. d)* 3^10 ÷ 3^5 ÷ 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 × 3 × 3 = 27 The value is 27. e)* 10^2 × 10^1 × 10^0 - 10^3 Step 1: Simplify 10^2 × 10^1 × 10^0 using the multiplication rule for exponents, a^m × a^n = a^m+n. 10^2+1+0 = 10^3 Step 2: Subtract 10^3. 10^3 - 10^3 = 0 The value is 0. f)* $3a^2 \