Simplify This Power Raised To A Power: $\left(5 4\right) 4$1. Expand Using Four Factors Of 5 4 5^4 5 4 : $5^4 \cdot 5^4 \cdot 5^4 \cdot 5^4$2. Apply The Product Of Powers Rule: $5^{16}$3. Simplify:

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Introduction

In mathematics, we often encounter expressions that involve powers raised to powers. These types of expressions can be simplified using the product of powers rule, which states that when we multiply two powers with the same base, we can add their exponents. In this article, we will explore how to simplify the expression $\left(5^4\right)^4$ using this rule.

Step 1: Expand Using Four Factors of $5^4$

To simplify the expression $\left(5^4\right)^4$, we can start by expanding it using four factors of $5^4$. This means that we can write the expression as:

$5^4 \cdot 5^4 \cdot 5^4 \cdot 5^4$

This expansion allows us to see the expression as a product of four powers of 5.

Step 2: Apply the Product of Powers Rule

Now that we have expanded the expression, we can apply the product of powers rule to simplify it. This rule states that when we multiply two powers with the same base, we can add their exponents. In this case, we have four powers of 5, each with an exponent of 4. We can add these exponents together to get:

$5^{4+4+4+4}$

This simplifies to:

$5^{16}$

Step 3: Simplify

We have now simplified the expression $\left(5^4\right)^4$ to $5^{16}$. This is the final answer.

Discussion

The product of powers rule is a powerful tool for simplifying expressions that involve powers raised to powers. By applying this rule, we can simplify complex expressions and make them easier to work with. In this case, we were able to simplify the expression $\left(5^4\right)^4$ to $5^{16}$ using the product of powers rule.

Real-World Applications

The product of powers rule has many real-world applications. For example, it can be used to simplify expressions that involve scientific notation. Scientific notation is a way of writing very large or very small numbers in a more manageable form. By applying the product of powers rule, we can simplify expressions that involve scientific notation and make them easier to work with.

Conclusion

In conclusion, the product of powers rule is a powerful tool for simplifying expressions that involve powers raised to powers. By applying this rule, we can simplify complex expressions and make them easier to work with. In this article, we explored how to simplify the expression $\left(5^4\right)^4$ using the product of powers rule. We also discussed the real-world applications of this rule and how it can be used to simplify expressions that involve scientific notation.

Additional Examples

Here are a few additional examples of how to simplify expressions using the product of powers rule:

• $\left(2^3\right)^5 = 2^{3+5} = 2^8$
• $\left(3^2\right)^4 = 3^{2+4} = 3^6$
• $\left(4^5\right)^3 = 4^{5+3} = 4^8$

These examples demonstrate how the product of powers rule can be used to simplify complex expressions and make them easier to work with.

Final Thoughts

Q: What is the product of powers rule?

A: The product of powers rule is a mathematical rule that states that when we multiply two powers with the same base, we can add their exponents. This rule is often represented as:

$a^m \cdot a^n = a^{m+n}$

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, simply add the exponents of the two powers. For example, if we have:

$2^3 \cdot 2^4$

We can apply the product of powers rule by adding the exponents:

$2^{3+4} = 2^7$

Q: What if the bases are different?

A: If the bases are different, we cannot apply the product of powers rule. For example, if we have:

$2^3 \cdot 3^4$

We cannot add the exponents because the bases are different. In this case, we would need to multiply the two numbers as is:

$2^3 \cdot 3^4 = 8 \cdot 81 = 648$

Q: Can I apply the product of powers rule to negative exponents?

A: Yes, we can apply the product of powers rule to negative exponents. For example, if we have:

$2^{-3} \cdot 2^{-4}$

We can add the exponents:

$2^{-3-4} = 2^{-7}$

Q: What if I have a power raised to a power?

A: If you have a power raised to a power, you can apply the product of powers rule by adding the exponents. For example, if we have:

$\left(2^3\right)^4$

We can apply the product of powers rule by adding the exponents:

$2^{3+4} = 2^7$

Q: Can I apply the product of powers rule to fractions?

A: Yes, we can apply the product of powers rule to fractions. For example, if we have:

$\left(\frac{1}{2}\right)^3 \cdot \left(\frac{1}{2}\right)^4$

We can add the exponents:

$\left(\frac{1}{2}\right)^{3+4} = \left(\frac{1}{2}\right)^7$

Q: What if I have a power with a variable base?

A: If you have a power with a variable base, you can apply the product of powers rule by adding the exponents. For example, if we have:

$x^3 \cdot x^4$

We can add the exponents:

$x^{3+4} = x^7$

Q: Can I apply the product of powers rule to complex numbers?

A: Yes, we can apply the product of powers rule to complex numbers. For example, if we have:

$\left(2+3i\right)^3 \cdot \left(2+3i\right)^4$

We can add the exponents:

$\left(2+3i\right)^{3+4} = \left(2+3i\right)^7$

Conclusion

The product of powers rule is a powerful tool for simplifying expressions that involve powers raised to powers. By applying this rule, we can simplify complex expressions and make them easier to work with. Whether you are working with integers, fractions, or complex numbers, the product of powers rule is a valuable tool to have in your mathematical toolkit.