Solve: ( 1 8 ) − 3 A = 512 3 A \left(\frac{1}{8}\right)^{-3a} = 512^{3a} ( 8 1 ​ ) − 3 A = 51 2 3 A A. A = − 8 A = -8 A = − 8 B. A = 0 A = 0 A = 0 C. A = 8 A = 8 A = 8 D. No Solution

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Understanding the Problem

The given equation involves exponents and powers of numbers. To solve for the variable $a$, we need to manipulate the equation using exponent rules and properties. The equation is $\left(\frac{1}{8}\right)^{-3a} = 512^{3a}$. Our goal is to isolate the variable $a$ and find its value.

Simplifying the Equation

We can start by simplifying the left-hand side of the equation. We know that $\frac{1}{8}$ can be written as $2^{-3}$. Therefore, we can rewrite the left-hand side as $(2^{-3})^{-3a}$.

Applying Exponent Rules

Using the rule for negative exponents, we can rewrite the left-hand side as $2^{9a}$. Now, the equation becomes $2^{9a} = 512^{3a}$.

Simplifying the Right-Hand Side

We can simplify the right-hand side by expressing $512$ as a power of $2$. Since $512 = 2^9$, we can rewrite the right-hand side as $(2^9)^{3a}$.

Applying Exponent Rules Again

Using the rule for exponents, we can rewrite the right-hand side as $2^{27a}$. Now, the equation becomes $2^{9a} = 2^{27a}$.

Equating Exponents

Since the bases are the same, we can equate the exponents. This gives us the equation $9a = 27a$.

Solving for $a$

To solve for $a$, we can subtract $9a$ from both sides of the equation. This gives us $0 = 18a$. Dividing both sides by $18$, we get $a = 0$.

Conclusion

Therefore, the value of $a$ that satisfies the equation is $a = 0$. This is the only solution to the equation.

Alternative Solution

We can also solve the equation by taking the logarithm of both sides. This will allow us to use logarithmic properties to simplify the equation and solve for $a$.

Taking the Logarithm

Taking the logarithm of both sides of the equation, we get $\log(2^{9a}) = \log(2^{27a})$.

Applying Logarithmic Properties

Using the property of logarithms that states $\log(a^b) = b\log(a)$, we can rewrite the equation as $9a\log(2) = 27a\log(2)$.

Canceling Out the Logarithm

Since the logarithm is the same on both sides, we can cancel it out. This gives us $9a = 27a$.

Solving for $a$

To solve for $a$, we can subtract $9a$ from both sides of the equation. This gives us $0 = 18a$. Dividing both sides by $18$, we get $a = 0$.

Conclusion

Therefore, the value of $a$ that satisfies the equation is $a = 0$. This is the only solution to the equation.

Final Answer

The final answer is $\boxed{0}$.

Discussion

The given equation involves exponents and powers of numbers. To solve for the variable $a$, we need to manipulate the equation using exponent rules and properties. The equation is $\left(\frac{1}{8}\right)^{-3a} = 512^{3a}$. Our goal is to isolate the variable $a$ and find its value.

Step 1: Simplify the Left-Hand Side

We can start by simplifying the left-hand side of the equation. We know that $\frac{1}{8}$ can be written as $2^{-3}$. Therefore, we can rewrite the left-hand side as $(2^{-3})^{-3a}$.

Step 2: Apply Exponent Rules

Using the rule for negative exponents, we can rewrite the left-hand side as $2^{9a}$. Now, the equation becomes $2^{9a} = 512^{3a}$.

Step 3: Simplify the Right-Hand Side

We can simplify the right-hand side by expressing $512$ as a power of $2$. Since $512 = 2^9$, we can rewrite the right-hand side as $(2^9)^{3a}$.

Step 4: Apply Exponent Rules Again

Using the rule for exponents, we can rewrite the right-hand side as $2^{27a}$. Now, the equation becomes $2^{9a} = 2^{27a}$.

Step 5: Equate Exponents

Since the bases are the same, we can equate the exponents. This gives us the equation $9a = 27a$.

Step 6: Solve for $a$

To solve for $a$, we can subtract $9a$ from both sides of the equation. This gives us $0 = 18a$. Dividing both sides by $18$, we get $a = 0$.

Step 7: Conclusion

Therefore, the value of $a$ that satisfies the equation is $a = 0$. This is the only solution to the equation.

Step 8: Alternative Solution

We can also solve the equation by taking the logarithm of both sides. This will allow us to use logarithmic properties to simplify the equation and solve for $a$.

Step 9: Taking the Logarithm

Taking the logarithm of both sides of the equation, we get $\log(2^{9a}) = \log(2^{27a})$.

Step 10: Applying Logarithmic Properties

Using the property of logarithms that states $\log(a^b) = b\log(a)$, we can rewrite the equation as $9a\log(2) = 27a\log(2)$.

Step 11: Canceling Out the Logarithm

Since the logarithm is the same on both sides, we can cancel it out. This gives us $9a = 27a$.

Step 12: Solving for $a$

To solve for $a$, we can subtract $9a$ from both sides of the equation. This gives us $0 = 18a$. Dividing both sides by $18$, we get $a = 0$.

Step 13: Conclusion

Therefore, the value of $a$ that satisfies the equation is $a = 0$. This is the only solution to the equation.

Step 14: Final Answer

The final answer is $\boxed{0}$.

Step 15: Discussion

The given equation involves exponents and powers of numbers. To solve for the variable $a$, we need to manipulate the equation using exponent rules and properties. The equation is $\left(\frac{1}{8}\right)^{-3a} = 512^{3a}$. Our goal is to isolate the variable $a$ and find its value.

Step 16: Step 1: Simplify the Left-Hand Side

We can start by simplifying the left-hand side of the equation. We know that $\frac{1}{8}$ can be written as $2^{-3}$. Therefore, we can rewrite the left-hand side as $(2^{-3})^{-3a}$.

Step 17: Step 2: Apply Exponent Rules

Using the rule for negative exponents, we can rewrite the left-hand side as $2^{9a}$. Now, the equation becomes $2^{9a} = 512^{3a}$.

Step 18: Step 3: Simplify the Right-Hand Side

We can simplify the right-hand side by expressing $512$ as a power of $2$. Since $512 = 2^9$, we can rewrite the right-hand side as $(2^9)^{3a}$.

Step 19: Step 4: Apply Exponent Rules Again

Using the rule for exponents, we can rewrite the right-hand side as $2^{27a}$. Now, the equation becomes $2^{9a} = 2^{27a}$.

Step 20: Step 5: Equate Exponents

Since the bases are the same, we can equate the exponents. This gives us the equation $9a = 27a$.

Step 21: Step 6: Solve for $a$

To solve for $a$, we can subtract $9a$ from both sides of the equation. This gives us $0 = 18a$. Dividing both sides by $18$, we get $a = 0$.

Step 22: Step 7: Conclusion

Therefore, the value of $a$ that satisfies the equation is $a = 0$. This is the only solution to the equation.

Step 23: Step 8: Alternative Solution

We can also solve the equation by taking the logarithm of both sides. This will allow us to use logarithmic properties to simplify the equation and solve for $a$.

Step 24: Step 9: Taking the Logarithm

Taking the logarithm of both sides of the equation, we get $\log(2^{9a}) = \log(2^{27a})$.

Step 25: Step 10: Applying Logarithmic Properties

Using the property of logarithms that states $\log(a^b) = b\log(a)$, we can rewrite the equation as $9a\log(2) = 27a

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Q: What is the main goal of solving the equation $\left(\frac{1}{8}\right)^{-3a} = 512^{3a}$?

A: The main goal is to isolate the variable $a$ and find its value.

Q: How do we start solving the equation?

A: We start by simplifying the left-hand side of the equation by expressing $\frac{1}{8}$ as a power of $2$. This gives us $(2^{-3})^{-3a}$.

Q: What is the next step in solving the equation?

A: We apply the rule for negative exponents to rewrite the left-hand side as $2^{9a}$. Now, the equation becomes $2^{9a} = 512^{3a}$.

Q: How do we simplify the right-hand side of the equation?

A: We express $512$ as a power of $2$, which gives us $(2^9)^{3a}$. Then, we apply the rule for exponents to rewrite the right-hand side as $2^{27a}$.

Q: What is the result of simplifying the right-hand side of the equation?

A: The equation becomes $2^{9a} = 2^{27a}$.

Q: How do we solve for $a$?

A: Since the bases are the same, we can equate the exponents. This gives us the equation $9a = 27a$.

Q: What is the next step in solving for $a$?

A: We subtract $9a$ from both sides of the equation to get $0 = 18a$.

Q: How do we solve for $a$?

A: We divide both sides of the equation by $18$ to get $a = 0$.

Q: Is there an alternative solution to solving the equation?

A: Yes, we can also solve the equation by taking the logarithm of both sides. This will allow us to use logarithmic properties to simplify the equation and solve for $a$.

Q: What is the result of taking the logarithm of both sides of the equation?

A: We get $\log(2^{9a}) = \log(2^{27a})$.

Q: How do we simplify the logarithmic equation?

A: We apply the property of logarithms that states $\log(a^b) = b\log(a)$ to rewrite the equation as $9a\log(2) = 27a\log(2)$.

Q: What is the result of simplifying the logarithmic equation?

A: We get $9a = 27a$.

Q: How do we solve for $a$ using the logarithmic equation?

A: We subtract $9a$ from both sides of the equation to get $0 = 18a$. Then, we divide both sides by $18$ to get $a = 0$.

Q: What is the final answer to the equation?

A: The final answer is $a = 0$.

Q: Is there a discussion about the solution to the equation?

A: Yes, the solution to the equation involves manipulating the equation using exponent rules and properties. The equation is $\left(\frac{1}{8}\right)^{-3a} = 512^{3a}$. Our goal is to isolate the variable $a$ and find its value.

Q: What are the steps involved in solving the equation?

A: The steps involved in solving the equation are:

1. Simplify the left-hand side of the equation by expressing $\frac{1}{8}$ as a power of $2$.
2. Apply the rule for negative exponents to rewrite the left-hand side as $2^{9a}$.
3. Simplify the right-hand side of the equation by expressing $512$ as a power of $2$.
4. Apply the rule for exponents to rewrite the right-hand side as $2^{27a}$.
5. Equate the exponents since the bases are the same.
6. Solve for $a$ by subtracting $9a$ from both sides of the equation and dividing both sides by $18$.

Q: What is the conclusion of the solution to the equation?

A: The value of $a$ that satisfies the equation is $a = 0$. This is the only solution to the equation.

Q: What is the final answer to the equation?

A: The final answer is $\boxed{0}$.