Let's solve each problem step by step using the rules of exponents. Here are the key rules we will use:
1.
Any number raised to the power of 0 is 1: \( a^0 = 1 \) (for \( a \neq 0 \)).
2.
Power of a power rule: \( (a^m)^n = a^{m \cdot n} \).
3.
Negative exponent rule: \( a^{-n} = \frac{1}{a^n} \).
Problem 4: \( (12^{-16})^0 \)
- Using the rule \( a^0 = 1 \):
\[
(12^{-16})^0 = 1
\]
-
Answer: \( \boxed{b} \)
Problem 5: \( ((10)^{-6})^{-2} \)
- Using the power of a power rule \( (a^m)^n = a^{m \cdot n} \):
\[
((10)^{-6})^{-2} = 10^{-6 \cdot -2} = 10^{12}
\]
-
Answer: \( \boxed{c} \)
Problem 6: \( ((8^4)^2)^3 \)
- Using the power of a power rule repeatedly:
\[
((8^4)^2)^3 = 8^{4 \cdot 2 \cdot 3} = 8^{24}
\]
-
Answer: \( \boxed{a} \)
Problem 7: \( \left( \left[ \frac{3}{4} \right]^2 \right)^6 \)
- Using the power of a power rule:
\[
\left( \left[ \frac{3}{4} \right]^2 \right)^6 = \left[ \frac{3}{4} \right]^{2 \cdot 6} = \left[ \frac{3}{4} \right]^{12}
\]
-
Answer: \( \boxed{d} \)
Problem 8: \( \left( \left( \frac{1}{9} \right)^{21} \right)^0 \)
- Using the rule \( a^0 = 1 \):
\[
\left( \left( \frac{1}{9} \right)^{21} \right)^0 = 1
\]
-
Answer: \( \boxed{b} \)
Problem 9: \( \left( \left( \frac{1}{4} \right)^{-14} \right)^2 \)
- Using the power of a power rule:
\[
\left( \left( \frac{1}{4} \right)^{-14} \right)^2 = \left( \frac{1}{4} \right)^{-14 \cdot 2} = \left( \frac{1}{4} \right)^{-28}
\]
-
Answer: \( \boxed{c} \)
Problem 10: \( \left( \left( \frac{5}{8} \right)^{-12} \right)^{-3} \)
- Using the power of a power rule:
\[
\left( \left( \frac{5}{8} \right)^{-12} \right)^{-3} = \left( \frac{5}{8} \right)^{-12 \cdot -3} = \left( \frac{5}{8} \right)^{36}
\]
-
Answer: \( \boxed{a} \)
Final Answers:
1. \( \boxed{b} \)
2. \( \boxed{c} \)
3. \( \boxed{a} \)
4. \( \boxed{d} \)
5. \( \boxed{b} \)
6. \( \boxed{c} \)
7. \( \boxed{a} \)
Parent Tip: Review the logic above to help your child master the concept of power to a power worksheet.
