Question

c) $9^6 \div 9^2$
$= 3^3$
e) $(4^4)^1$
$= 4^4$
g) $(s^3t^5)^4$
$= s^{24}t^{3125}$

Explanation:

Part c)

Step1: Apply exponent rule \(a^m \div a^n = a^{m - n}\)

For \(9^6 \div 9^2\), here \(a = 9\), \(m = 6\), \(n = 2\). So we have \(9^{6 - 2}\).

Step2: Simplify the exponent

\(6 - 2 = 4\), so \(9^4\). If we want to express it as a power of 3, since \(9=3^2\), then \((3^2)^4=3^{2\times4}=3^8\), but the original wrong answer was \(3^3\), the correct simplification using the quotient rule of exponents is \(9^{6 - 2}=9^4 = 6561\) or as \(3^8\). But following the exponent rule for division: \(a^m\div a^n=a^{m - n}\), so \(9^6\div9^2 = 9^{6 - 2}=9^4\) or \(3^8\).

Part e)

Step1: Apply exponent rule \((a^m)^n=a^{m\times n}\)

For \((4^4)^1\), here \(a = 4\), \(m = 4\), \(n = 1\). So we have \(4^{4\times1}\).

Step2: Simplify the exponent

\(4\times1 = 4\), so \((4^4)^1=4^4 = 256\). The given answer \(4^4\) is correct as \((a^m)^n=a^{m\times n}\) and when \(n = 1\), \((a^m)^1=a^m\).

Part g)

Step1: Apply exponent rule \((ab)^n=a^n\times b^n\) and \((a^m)^n=a^{m\times n}\)

For \((s^3t^5)^4\), first apply \((ab)^n=a^n b^n\) to get \((s^3)^4\times(t^5)^4\). Then apply \((a^m)^n=a^{m\times n}\) to each term: \((s^3)^4=s^{3\times4}=s^{12}\) and \((t^5)^4=t^{5\times4}=t^{20}\). So \((s^3t^5)^4=s^{12}t^{20}\). The original answer has errors, the correct simplification using power of a product and power of a power rules is \(s^{12}t^{20}\).

Final Answers:

• c) Correct result: \(9^4\) (or \(3^8\) or \(6561\))
• e) Correct result: \(4^4\) (or \(256\))
• g) Correct result: \(s^{12}t^{20}\)

Answer:

Step1: Apply exponent rule \((ab)^n=a^n\times b^n\) and \((a^m)^n=a^{m\times n}\)

For \((s^3t^5)^4\), first apply \((ab)^n=a^n b^n\) to get \((s^3)^4\times(t^5)^4\). Then apply \((a^m)^n=a^{m\times n}\) to each term: \((s^3)^4=s^{3\times4}=s^{12}\) and \((t^5)^4=t^{5\times4}=t^{20}\). So \((s^3t^5)^4=s^{12}t^{20}\). The original answer has errors, the correct simplification using power of a product and power of a power rules is \(s^{12}t^{20}\).

Final Answers:

• c) Correct result: \(9^4\) (or \(3^8\) or \(6561\))
• e) Correct result: \(4^4\) (or \(256\))
• g) Correct result: \(s^{12}t^{20}\)