Simplify $\left(8 M^5\right)^2$.

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Understanding the Problem

When simplifying an expression like $\left(8 m^5\right)^2$, we need to apply the rules of exponents to simplify the expression. The expression involves a coefficient and a variable raised to a power, which is then squared.

Applying the Rules of Exponents

To simplify the expression, we need to apply the rule of exponents that states $(ab)^n = a^n b^n$. In this case, we have $(8 m^5)^2$, which can be rewritten as $8^2 (m^5)^2$.

Simplifying the Coefficient

The coefficient $8$ is raised to the power of $2$, which means we need to multiply $8$ by itself $2$ times. This gives us $8^2 = 64$.

Simplifying the Variable

The variable $m$ is raised to the power of $5$, which is then squared. This means we need to multiply $m^5$ by itself $2$ times. Using the rule of exponents that states $(a^m)^n = a^{mn}$, we can simplify this to $m^{5 \cdot 2} = m^{10}$.

Combining the Simplified Coefficient and Variable

Now that we have simplified the coefficient and the variable, we can combine them to get the final simplified expression. This gives us $64 m^{10}$.

Conclusion

In conclusion, simplifying the expression $\left(8 m^5\right)^2$ involves applying the rules of exponents to simplify the coefficient and the variable. By following these rules, we can simplify the expression to $64 m^{10}$.

Example Use Case

Simplifying expressions like $\left(8 m^5\right)^2$ is an important skill in mathematics, particularly in algebra and calculus. It is used to simplify complex expressions and make them easier to work with.

Tips and Tricks

When simplifying expressions like $\left(8 m^5\right)^2$, it is essential to follow the rules of exponents carefully. This includes applying the rule that $(ab)^n = a^n b^n$ and the rule that $(a^m)^n = a^{mn}$.

Common Mistakes

One common mistake when simplifying expressions like $\left(8 m^5\right)^2$ is to forget to apply the rules of exponents. This can lead to incorrect simplifications and make it difficult to work with the expression.

Real-World Applications

Simplifying expressions like $\left(8 m^5\right)^2$ has many real-world applications, particularly in science and engineering. It is used to simplify complex equations and make them easier to solve.

Final Thoughts

In conclusion, simplifying the expression $\left(8 m^5\right)^2$ involves applying the rules of exponents to simplify the coefficient and the variable. By following these rules, we can simplify the expression to $64 m^{10}$.

Frequently Asked Questions

• Q: What is the simplified form of $\left(8 m^5\right)^2$? A: The simplified form of $\left(8 m^5\right)^2$ is $64 m^{10}$.
• Q: How do I simplify expressions like $\left(8 m^5\right)^2$? A: To simplify expressions like $\left(8 m^5\right)^2$, you need to apply the rules of exponents, including the rule that $(ab)^n = a^n b^n$ and the rule that $(a^m)^n = a^{mn}$.

Further Reading

For more information on simplifying expressions like $\left(8 m^5\right)^2$, check out the following resources:

• Algebra Rules of Exponents
• Simplifying Expressions
• Exponents and Powers
  

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Frequently Asked Questions

Q: What is the simplified form of $\left(8 m^5\right)^2$?

A: The simplified form of $\left(8 m^5\right)^2$ is $64 m^{10}$.

Q: How do I simplify expressions like $\left(8 m^5\right)^2$?

A: To simplify expressions like $\left(8 m^5\right)^2$, you need to apply the rules of exponents, including the rule that $(ab)^n = a^n b^n$ and the rule that $(a^m)^n = a^{mn}$.

Q: What is the rule for simplifying expressions with exponents?

A: The rule for simplifying expressions with exponents is to multiply the coefficients and add the exponents. For example, $(ab)^n = a^n b^n$ and $(a^m)^n = a^{mn}$.

Q: How do I handle negative exponents?

A: To handle negative exponents, you need to rewrite the expression with a positive exponent. For example, $a^{-n} = \frac{1}{a^n}$.

Q: Can I simplify expressions with fractional exponents?

A: Yes, you can simplify expressions with fractional exponents. For example, $(a^m)^n = a^{mn}$ and $(a^m)^{\frac{1}{n}} = a^{\frac{m}{n}}$.

Q: How do I simplify expressions with radicals?

A: To simplify expressions with radicals, you need to apply the rules of radicals, including the rule that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.

Q: Can I simplify expressions with absolute values?

A: Yes, you can simplify expressions with absolute values. For example, $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$.

Q: How do I simplify expressions with complex numbers?

A: To simplify expressions with complex numbers, you need to apply the rules of complex numbers, including the rule that $i^2 = -1$.

Q: Can I simplify expressions with trigonometric functions?

A: Yes, you can simplify expressions with trigonometric functions. For example, $\sin^2 x + \cos^2 x = 1$.

Q: How do I simplify expressions with logarithmic functions?

A: To simplify expressions with logarithmic functions, you need to apply the rules of logarithms, including the rule that $\log_a b = \frac{\log_c b}{\log_c a}$.

Q: Can I simplify expressions with exponential functions?

A: Yes, you can simplify expressions with exponential functions. For example, $e^x \cdot e^y = e^{x+y}$.

Q: How do I simplify expressions with polynomial functions?

A: To simplify expressions with polynomial functions, you need to apply the rules of polynomial functions, including the rule that $(a+b)^n = a^n + na^{n-1}b + \ldots + b^n$.

Additional Resources

For more information on simplifying expressions, check out the following resources:

• Algebra Rules of Exponents
• Simplifying Expressions
• Exponents and Powers
• Radicals and Roots
• Absolute Values
• Complex Numbers
• Trigonometric Functions
• Logarithmic Functions
• Exponential Functions
• Polynomial Functions

Conclusion

Simplifying expressions is an essential skill in mathematics, particularly in algebra and calculus. By applying the rules of exponents, radicals, and other mathematical operations, you can simplify complex expressions and make them easier to work with. Remember to always follow the rules of mathematics and to check your work carefully to ensure that your answers are correct.