The main goal is to apply the exponent rules to simplify the expression

Oct 13, 2025 by magicalceylontours.com 116 views

Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. The expression given involves multiplying two terms with exponents, and we need to simplify it using the properties of exponents. The main goal is to apply the exponent rules to simplify the expression.

Applying Exponent Rules

To simplify the given expression, we need to apply the exponent rules. The first rule we'll use is the power of a power rule, which states that $(a^m)^n = a^{mn}$ . We can apply this rule to both terms in the expression.

Power of a Power Rule

The power of a power rule allows us to simplify expressions like $(a^m)^n$ by multiplying the exponents. In this case, we have:

\left(2 x^4\right)^3 = 2^3 \cdot \left(x^4\right)^3 = 8 x^{12}

and

\left(5 x^7\right)^2 = 5^2 \cdot \left(x^7\right)^2 = 25 x^{14}

Multiplying Terms with Exponents

Now that we've simplified the individual terms, we can multiply them together. When multiplying terms with exponents, we add the exponents. In this case, we have:

11 \cdot \left(2 x^4\right)^3 \cdot \left(5 x^7\right)^2 = 11 \cdot 8 x^{12} \cdot 25 x^{14}

Simplifying the Expression

Now that we've multiplied the terms, we can simplify the expression by combining the constants and adding the exponents.

Combining Constants

The constants in the expression are 11, 8, and 25. We can combine these constants by multiplying them together:

11 \cdot 8 \cdot 25 = 2200

Adding Exponents

The exponents in the expression are $x^{12}$ and $x^{14}$ . We can add these exponents by adding the exponents:

x^{12} \cdot x^{14} = x^{12 + 14} = x^{26}

Final Simplified Expression

Now that we've combined the constants and added the exponents, we can write the final simplified expression:

2200 x^{26}

Conclusion

In this article, we've simplified the expression $11 \cdot \left(2 x^4\right)^3 \cdot \left(5 x^7\right)^2$ using the properties of exponents. We applied the power of a power rule to simplify the individual terms and then multiplied them together. Finally, we combined the constants and added the exponents to write the final simplified expression.

Frequently Asked Questions

What is the power of a power rule? The power of a power rule states that $(a^m)^n = a^{mn}$ .
How do you multiply terms with exponents? When multiplying terms with exponents, you add the exponents.
What is the final simplified expression? The final simplified expression is $2200 x^{26}$ .

References

[1] Khan Academy. (n.d.). Exponent Rules. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7c/x2f6f7d/x2f6f7e
[2] Mathway. (n.d.). Exponent Rules. Retrieved from https://www.mathway.com/subjects/exponent-rules

Frequently Asked Questions

Q: What is the power of a power rule?

A: The power of a power rule states that $(a^m)^n = a^{mn}$ . This means that when you have an exponent raised to another exponent, you multiply the exponents.

Q: How do you multiply terms with exponents?

A: When multiplying terms with exponents, you add the exponents. For example, if you have $x^2 \cdot x^3$ , you would add the exponents to get $x^{2+3} = x^5$ .

Q: What is the final simplified expression?

A: The final simplified expression is $2200 x^{26}$ .

Q: Can you explain the steps to simplify the expression?

A: Yes, here are the steps to simplify the expression:

Apply the power of a power rule to each term: $\left(2 x^4\right)^3 = 2^3 \cdot \left(x^4\right)^3 = 8 x^{12}$ and $\left(5 x^7\right)^2 = 5^2 \cdot \left(x^7\right)^2 = 25 x^{14}$ .
Multiply the terms together: $11 \cdot 8 x^{12} \cdot 25 x^{14}$ .
Combine the constants: $11 \cdot 8 \cdot 25 = 2200$ .
Add the exponents: $x^{12} \cdot x^{14} = x^{12 + 14} = x^{26}$ .
Write the final simplified expression: $2200 x^{26}$ .

Q: What are some common mistakes to avoid when simplifying expressions with exponents?

A: Some common mistakes to avoid when simplifying expressions with exponents include:

Forgetting to apply the power of a power rule
Not adding the exponents when multiplying terms with exponents
Not combining the constants
Not writing the final simplified expression in the correct form

Q: Can you provide more examples of simplifying expressions with exponents?

A: Yes, here are a few more examples:

Simplify the expression $3 \cdot \left(2 x^2\right)^4 \cdot \left(5 x^3\right)^2$ .
Simplify the expression $2 \cdot \left(3 x^4\right)^3 \cdot \left(4 x^5\right)^2$ .
Simplify the expression $5 \cdot \left(2 x^2\right)^3 \cdot \left(3 x^4\right)^2$ .

Additional Resources

Exponent Rules: A Comprehensive Guide
Simplifying Expressions with Exponents
Multiplying Terms with Exponents

References

[1] Khan Academy. (n.d.). Exponent Rules. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7c/x2f6f7d/x2f6f7e
[2] Mathway. (n.d.). Exponent Rules. Retrieved from https://www.mathway.com/subjects/exponent-rules