Question

reteaching 8-4
objective: using two more multiplication properties of exponents
materials: none

• to raise a power to a power, multiply the exponents
• every number and variable inside parentheses is being raised to the power to the right of the parentheses.

example
simplify ((4x^3)^2)
((4x^3)^2)
((4^1x^3)^2) —— rewrite each number and variable with an exponent.
((4^1x^3)^2) —— draw arrows from the exponent outside the parentheses to each exponent inside the parentheses.
(4^{2}x^{2cdot3}) —— rewrite, showing the exponents to be multiplied.
(4^2x^6) —— multiply the exponents.
(16x^6) —— simplify.
exercises
draw arrows from the exponent outside the parentheses to each exponent inside the parentheses. then simplify each expression.

1. ((5^3)^4) 2. ((a^6)^4) 3. ((2^3)^2) 4. ((4x^3)^2)
2. ((3a^4)^2) 6. ((2g^2)^3) 7. ((g^2h^3)^3) 8. ((x^6)^2)

simplify each expression.

1. ((a^2b^4)^3) 10. ((3x^5)^0) 11. (g^9cdot g^{-3})
2. ((c^4)^7) 13. ((3.2)^5cdot(3.2)^{-4}) 14. ((8ab^6)^3)
3. ((a^3b^3)^2) 16. ((x^3)^2) 17. ((3x^2y)^2)
4. ((-2x^3)^3) 19. ((x^3y^4)^3) 20. ((3x^2y)^3)
5. ((-4x^2y^3)^3) 22. ((xyz)^0) 23. (r^5cdot r^{-7})

Explanation:

Let's solve one of the problems, say problem 1: Simplify \((5^3)^4\)

Step1: Identify the exponent rule

To raise a power to a power, we multiply the exponents. The formula for \((a^m)^n\) is \(a^{m\times n}\). Here, \(a = 5\), \(m = 3\), and \(n = 4\).

Step2: Apply the exponent rule

Multiply the exponents: \(3\times4 = 12\). So, \((5^3)^4=5^{3\times4}=5^{12}\)

Step3: Calculate \(5^{12}\) (optional, but if needed)

\(5^{12}=5\times5\times5\times5\times5\times5\times5\times5\times5\times5\times5\times5 = 244140625\) (if we want the numerical value)

Step1: Use the power - of - a - power rule

The rule for \((a^m)^n\) is \(a^{m\times n}\). Here, \(a=a\), \(m = 6\), \(n = 4\)

Step2: Multiply the exponents

\(6\times4=24\), so \((a^6)^4=a^{6\times4}=a^{24}\)

Step1: Apply the power - of - a - power rule

For \((a^m)^n\), we have \(a^{m\times n}\). Here, \(a = 2\), \(m = 3\), \(n = 2\)

Step2: Multiply the exponents

\(3\times2 = 6\), so \((2^3)^2=2^{3\times2}=2^6\)

Step3: Calculate \(2^6\)

\(2^6=2\times2\times2\times2\times2\times2 = 64\)

Answer:

If we leave it in exponential form, the answer is \(5^{12}\). If we calculate the numerical value, the answer is \(244140625\)

Let's solve another one, problem 2: Simplify \((a^6)^4\)