complete the chart and then determine the rule for the power of powers.\nproblem | expansion |…

complete the chart and then determine the rule for the power of powers.\nproblem | expansion | solution\n$(5^3)^4$ | $5^3(5^3)(5^3)(5^3) = (5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)$ | $5^{12}$\n$((-3)^4)^2$ | $(-3)^4(-3)^4 = (-3)(-3)(-3)(-3)(-3)(-3)(-3)(-3)$ | $(-3)^8$\n$(2^5)^3$ | $(2)^5(2)^5(2)^5 = (2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)$ | \n$(x^7)^2$ | | $x^{14}$\n$(y^4)^5$ | | \nwrite the rule: $(b^m)^n = $\nuse this rule to simplify each expression.\n$(8^6)^4 = $ | $((-5)^9)^3 = $ | $(x^6)^7 = $ | $(y^{-4})^{-5} = $

complete the chart and then determine the rule for the power of powers.\nproblem | expansion | solution\n$(5^3)^4$ | $5^3(5^3)(5^3)(5^3) = (5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)$ | $5^{12}$\n$((-3)^4)^2$ | $(-3)^4(-3)^4 = (-3)(-3)(-3)(-3)(-3)(-3)(-3)(-3)$ | $(-3)^8$\n$(2^5)^3$ | $(2)^5(2)^5(2)^5 = (2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)$ | \n$(x^7)^2$ | | $x^{14}$\n$(y^4)^5$ | | \nwrite the rule: $(b^m)^n = $\nuse this rule to simplify each expression.\n$(8^6)^4 = $ | $((-5)^9)^3 = $ | $(x^6)^7 = $ | $(y^{-4})^{-5} = $

Answer

Explanation:

Step1: Solve ((2^5)^3)

Count the number of 2s in the expansion: there are (5\times3 = 15) twos, so ((2^5)^3=2^{15}).

Step2: Expand ((x^7)^2)

((x^7)^2=x^7\times x^7=(x\times x\times x\times x\times x\times x\times x)\times(x\times x\times x\times x\times x\times x\times x)=x^{14}) (matches the given solution).

Step3: Expand and solve ((y^4)^5)

Expansion: ((y^4)^5 = y^4\times y^4\times y^4\times y^4\times y^4=(y\times y\times y\times y)\times(y\times y\times y\times y)\times(y\times y\times y\times y)\times(y\times y\times y\times y)\times(y\times y\times y\times y)). Number of (y)s: (4\times5 = 20), so solution is (y^{20}).

Step4: Determine the Power of Powers rule

From the problems: ((5^3)^4 = 5^{3\times4}=5^{12}), ((( - 3)^4)^2=(-3)^{4\times2}=(-3)^8), ((2^5)^3 = 2^{5\times3}=2^{15}), ((x^7)^2=x^{7\times2}=x^{14}), ((y^4)^5=y^{4\times5}=y^{20}). So the rule is ((b^m)^n=b^{m\times n}=b^{mn}).

Step5: Simplify using the rule

  • ((8^6)^4): Using ((b^m)^n = b^{mn}), (m = 6), (n = 4), so (8^{6\times4}=8^{24}).
  • ((( - 5)^9)^3): (m = 9), (n = 3), so ((-5)^{9\times3}=(-5)^{27}).
  • ((x^6)^7): (m = 6), (n = 7), so (x^{6\times7}=x^{42}).
  • ((y^{-4})^{-5}): (m=-4), (n = - 5), so (y^{(-4)\times(-5)}=y^{20}).

Answer:

  • ((2^5)^3) Solution: (2^{15})
  • ((x^7)^2) Expansion: (x^7\times x^7=(x\times x\times x\times x\times x\times x\times x)\times(x\times x\times x\times x\times x\times x\times x))
  • ((y^4)^5) Expansion: (y^4\times y^4\times y^4\times y^4\times y^4=(y\times y\times y\times y)\times(y\times y\times y\times y)\times(y\times y\times y\times y)\times(y\times y\times y\times y)\times(y\times y\times y\times y)); Solution: (y^{20})
  • Power of Powers Rule: ((b^m)^n = b^{mn})
  • ((8^6)^4 = 8^{24})
  • ((( - 5)^9)^3=(-5)^{27})
  • ((x^6)^7 = x^{42})
  • ((y^{-4})^{-5}=y^{20})