Simplify The Expression:${ 5^4 \times 5^6 = }$A. { 25^{10} $}$B. { 10^{10} $}$C. { 5^{10} $}$D. { 5^{24} $}$

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Understanding Exponents and Their Rules

When dealing with exponents, it's essential to understand the rules that govern their behavior. Exponents are a shorthand way of representing repeated multiplication. For example, $5^4$ means $5$ multiplied by itself $4$ times, which is equal to $5 \times 5 \times 5 \times 5 = 625$. The exponent $4$ tells us how many times to multiply the base number $5$.

The Product of Powers Rule

One of the fundamental rules of exponents is the product of powers rule, which states that when we multiply two numbers with the same base, we add their exponents. In other words, if we have $a^m \times a^n$, where $a$ is the base and $m$ and $n$ are the exponents, then the result is $a^{m+n}$. This rule can be applied to any base, not just $5$.

Applying the Product of Powers Rule to the Given Expression

Now, let's apply the product of powers rule to the given expression: $5^4 \times 5^6$. Since both terms have the same base, $5$, we can add their exponents. Therefore, $5^4 \times 5^6 = 5^{4+6} = 5^{10}$.

Evaluating the Result

The result of the expression is $5^{10}$. To evaluate this, we need to multiply the base number $5$ by itself $10$ times. However, we can also express this result in a different form using the fact that $5^2 = 25$. Therefore, $5^{10} = (5^2)^5 = 25^5$. However, this is not among the answer choices.

Alternative Form of the Result

Another way to express the result is to use the fact that $5^2 = 25$. Therefore, $5^{10} = (5^2)^5 = 25^5$. However, this is not among the answer choices. We can also express the result as $5^{10} = 5^2 \times 5^2 \times 5^2 \times 5^2 \times 5^2 = 25 \times 25 \times 25 \times 25 \times 25 = 25^5$. However, this is not among the answer choices.

Comparing the Result with the Answer Choices

Now, let's compare the result, $5^{10}$, with the answer choices. We can see that option C, $5^{10}$, is the correct answer. The other options are incorrect because they do not match the result of the expression.

Conclusion

In conclusion, the expression $5^4 \times 5^6$ can be simplified using the product of powers rule, which states that when we multiply two numbers with the same base, we add their exponents. Therefore, $5^4 \times 5^6 = 5^{4+6} = 5^{10}$. This result can be expressed in different forms, but the correct answer is option C, $5^{10}$.

Frequently Asked Questions

• What is the product of powers rule? The product of powers rule states that when we multiply two numbers with the same base, we add their exponents.
• How do we apply the product of powers rule to the given expression? We add the exponents of the two terms, $5^4$ and $5^6$, to get $5^{4+6} = 5^{10}$.
• What is the result of the expression $5^4 \times 5^6$? The result of the expression is $5^{10}$.

Key Takeaways

• The product of powers rule states that when we multiply two numbers with the same base, we add their exponents.
• We can apply the product of powers rule to simplify the expression $5^4 \times 5^6$.
• The result of the expression is $5^{10}$.

Further Reading

• Exponents and their rules
• Product of powers rule
• Simplifying expressions with exponents

References

• [1] Khan Academy. (n.d.). Exponents and Exponential Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7d/x2f5f7d/exponents-and-exponential-functions
• [2] Math Open Reference. (n.d.). Exponents. Retrieved from https://www.mathopenref.com/exponents.html

Related Topics

• Exponents and their rules
• Product of powers rule
• Simplifying expressions with exponents

Tags

• Exponents
• Product of powers rule
• Simplifying expressions
• Math
• Algebra
  

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Frequently Asked Questions

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two numbers with the same base, we add their exponents. In other words, if we have $a^m \times a^n$, where $a$ is the base and $m$ and $n$ are the exponents, then the result is $a^{m+n}$.

Q: How do we apply the product of powers rule to the given expression?

A: We add the exponents of the two terms, $5^4$ and $5^6$, to get $5^{4+6} = 5^{10}$.

Q: What is the result of the expression $5^4 \times 5^6$?

A: The result of the expression is $5^{10}$.

Q: Can we simplify the expression further?

A: No, the expression $5^{10}$ is already in its simplest form.

Q: How do we evaluate the result $5^{10}$?

A: To evaluate the result, we need to multiply the base number $5$ by itself $10$ times. However, we can also express this result in a different form using the fact that $5^2 = 25$. Therefore, $5^{10} = (5^2)^5 = 25^5$.

Q: Why is option C, $5^{10}$, the correct answer?

A: Option C, $5^{10}$, is the correct answer because it is the result of the expression $5^4 \times 5^6$.

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

A: Some common mistakes to avoid include:

• Not applying the product of powers rule when multiplying two numbers with the same base.
• Not adding the exponents when multiplying two numbers with the same base.
• Not simplifying the expression further when possible.

Q: How do we know when to apply the product of powers rule?

A: We apply the product of powers rule when we are multiplying two numbers with the same base.

Q: Can we apply the product of powers rule to expressions with different bases?

A: No, the product of powers rule only applies to expressions with the same base.

Q: What are some real-world applications of the product of powers rule?

A: The product of powers rule has many real-world applications, including:

• Calculating the area and volume of shapes.
• Modeling population growth and decay.
• Solving problems in physics and engineering.

Additional Resources

• Khan Academy: Exponents and Exponential Functions
• Math Open Reference: Exponents
• Wolfram MathWorld: Exponents and Exponential Functions

Related Topics

• Exponents and their rules
• Product of powers rule
• Simplifying expressions with exponents
• Calculus
• Algebra

Tags

• Exponents
• Product of powers rule
• Simplifying expressions
• Math
• Algebra
• Calculus