Simplify The Expression:$\left(6 \sqrt{64 Y^{12}}\right)^{-5}$

Introduction

In this article, we will simplify the given expression $\left(6 \sqrt{64 y^{12}}\right)^{-5}$. This involves applying the properties of exponents and radicals to simplify the expression. We will break down the process step by step, making it easy to understand and follow.

Step 1: Simplify the Radical

The expression contains a square root, which can be simplified using the properties of radicals. We can rewrite $\sqrt{64 y^{12}}$ as $64^{1/2} y^{12/2}$, which simplifies to $8y^6$.

import math

# Define the variables
a = 64
b = 12
c = 6

# Simplify the radical
radical = math.sqrt(a * (c ** b))
print(radical)

Step 2: Apply the Power Rule

Now that we have simplified the radical, we can apply the power rule to simplify the expression. The power rule states that for any non-zero number $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$. We can apply this rule to simplify the expression $\left(6 \sqrt{64 y^{12}}\right)^{-5}$.

# Define the variables
a = 6
b = 8
c = 6
d = -5

# Apply the power rule
expression = (a * b * (c ** d))
print(expression)

Step 3: Simplify the Expression

Now that we have applied the power rule, we can simplify the expression further. We can rewrite the expression as $6^{-5} \cdot 8^{-5} \cdot (y^6)^{-5}$.

# Define the variables
a = 6
b = 8
c = 6
d = -5

# Simplify the expression
expression = (a ** d) * (b ** d) * ((c ** d))
print(expression)

Step 4: Evaluate the Expression

Finally, we can evaluate the expression by simplifying the exponents. We can rewrite the expression as $\frac{1}{6^5} \cdot \frac{1}{8^5} \cdot \frac{1}{y^{30}}$.

# Define the variables
a = 6
b = 8
c = 30

# Evaluate the expression
expression = (1 / (a ** 5)) * (1 / (b ** 5)) * (1 / (c ** 1))
print(expression)

Conclusion

In this article, we simplified the expression $\left(6 \sqrt{64 y^{12}}\right)^{-5}$ using the properties of exponents and radicals. We broke down the process into four steps, making it easy to understand and follow. We applied the power rule to simplify the expression and evaluated the final result.

Final Answer

Introduction

In our previous article, we simplified the expression $\left(6 \sqrt{64 y^{12}}\right)^{-5}$ using the properties of exponents and radicals. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q: What is the property of exponents used in simplifying the expression?

A: The property of exponents used in simplifying the expression is the power rule, which states that for any non-zero number $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$.

Q: How do you simplify the radical in the expression?

A: To simplify the radical, we can rewrite $\sqrt{64 y^{12}}$ as $64^{1/2} y^{12/2}$, which simplifies to $8y^6$.

Q: What is the difference between the power rule and the product rule?

A: The power rule states that for any non-zero number $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$. The product rule states that for any non-zero numbers $a$ and $b$ and integers $m$ and $n$, $(ab)^m = a^mb^m$.

Q: How do you simplify the expression using the power rule?

A: To simplify the expression using the power rule, we can rewrite $\left(6 \sqrt{64 y^{12}}\right)^{-5}$ as $6^{-5} \cdot 8^{-5} \cdot (y^6)^{-5}$.

Q: What is the final answer to the expression?

A: The final answer to the expression is $\frac{1}{6^5} \cdot \frac{1}{8^5} \cdot \frac{1}{y^{30}}$.

Q: Can you provide a step-by-step solution to the expression?

A: Yes, here is a step-by-step solution to the expression:

1. Simplify the radical: $\sqrt{64 y^{12}} = 8y^6$
2. Apply the power rule: $\left(6 \sqrt{64 y^{12}}\right)^{-5} = 6^{-5} \cdot 8^{-5} \cdot (y^6)^{-5}$
3. Simplify the expression: $6^{-5} \cdot 8^{-5} \cdot (y^6)^{-5} = \frac{1}{6^5} \cdot \frac{1}{8^5} \cdot \frac{1}{y^{30}}$

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\left(6 \sqrt{64 y^{12}}\right)^{-5}$. We provided step-by-step solutions and explanations to help readers understand the process of simplifying the expression.

Final Answer

The final answer is $\boxed{\frac{1}{6^5} \cdot \frac{1}{8^5} \cdot \frac{1}{y^{30}}}$.