Sure! Below are the handwritten calculations for each of the problems you provided.
1. Simplify \( 6^{10} \times 6^{5} \) using the Product Rule of Exponents.
The Product Rule of Exponents states that when you multiply like bases, you add the exponents.
\[ 6^{10} \times 6^{5} = 6^{10 + 5} = 6^{15} \]
Answer:
\[ 6^{15} \]
2. Find the numerical equivalent of \( 8^{9} \times 8^{-6} \).
Using the Product Rule of Exponents, we start by simplifying:
\[ 8^{9} \times 8^{-6} = 8^{9 + (-6)} = 8^{9 - 6} = 8^{3} \]
Now, we find the numerical equivalent of \( 8^{3} \):
\[ 8^{3} = 512 \]
Answer:
\[ 512 \]
3. Find the value of \( x \) to make the equation \( \frac{5^{x}}{5^{5}} = 5^{12} \) true.
Using the Quotient Rule of Exponents, which states that \( \frac{a^m}{a^n} = a^{m-n} \):
\[ 5^{x - 5} = 5^{12} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ x - 5 = 12 \]
To solve for \( x \):
\[ x = 12 + 5 = 17 \]
Answer:
\[ x = 17 \]
4. Simplify \( (9^{3})^{6} \) using the Power Rule of Exponents.
The Power Rule of Exponents states that \( (a^{m})^{n} = a^{mn} \):
\[ (9^{3})^{6} = 9^{3 \times 6} = 9^{18} \]
Answer:
\[ 9^{18} \]
5. Simplify \( \frac{(6^{-3})^{7} \times 6^{20}}{6^{-2}} \).
Using the Power Rule on \( (6^{-3})^{7} \):
\[ (6^{-3})^{7} = 6^{-21} \]
Now, we can rewrite the expression:
\[ \frac{6^{-21} \times 6^{20}}{6^{-2}} \]
Using the Product Rule in the numerator:
\[ 6^{-21 + 20} = 6^{-1} \]
Now applying the Quotient Rule:
\[ \frac{6^{-1}}{6^{-2}} = 6^{-1 - (-2)} = 6^{-1 + 2} = 6^{1} \]
So the final simplified expression is:
Answer:
\[ 6^{1} = 6 \]
Or in improper fraction form:
\[ \frac{6}{1} \]
Feel free to let me know if you need further assistance!