Apply The Power Rule To Simplify The Expression:$\left(4^2\right)^5 =$

Introduction

In mathematics, the power rule is a fundamental concept used to simplify exponential expressions. It states that when raising a power to another power, we multiply the exponents. In this article, we will apply the power rule to simplify the expression $\left(4^2\right)^5$. We will explore the concept of exponents, the power rule, and provide step-by-step examples to demonstrate its application.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $4^2$ means $4$ multiplied by itself $2$ times, which is equal to $16$. Exponents can be thought of as a power or a quantity that is raised to a certain value. In the expression $\left(4^2\right)^5$, the exponent $2$ is being raised to the power of $5$.

The Power Rule

The power rule states that when raising a power to another power, we multiply the exponents. In mathematical notation, this can be represented as:

$(a^m)^n = a^{m \cdot n}$

where $a$ is the base, $m$ is the exponent, and $n$ is the power. Using this rule, we can simplify the expression $\left(4^2\right)^5$.

Applying the Power Rule

To simplify the expression $\left(4^2\right)^5$, we will apply the power rule by multiplying the exponents. The exponent $2$ is being raised to the power of $5$, so we multiply $2$ by $5$ to get $10$.

$\left(4^2\right)^5 = 4^{2 \cdot 5} = 4^{10}$

Simplifying the Expression

Now that we have simplified the expression to $4^{10}$, we can evaluate it by multiplying $4$ by itself $10$ times.

$4^{10} = 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 1,048,576$

Conclusion

In this article, we applied the power rule to simplify the expression $\left(4^2\right)^5$. We explored the concept of exponents, the power rule, and provided step-by-step examples to demonstrate its application. By understanding and applying the power rule, we can simplify complex exponential expressions and evaluate them to get their final values.

Real-World Applications

The power rule has numerous real-world applications in fields such as science, engineering, and finance. For example, in physics, the power rule is used to calculate the energy of a system, while in finance, it is used to calculate compound interest.

Common Mistakes

When applying the power rule, it is essential to remember that the exponents are multiplied, not added. For example, $\left(4^2\right)^5$ is not equal to $4^{2+5}$, but rather $4^{2 \cdot 5}$.

Practice Problems

To practice applying the power rule, try simplifying the following expressions:

• $\left(3^4\right)^2$
• $\left(2^3\right)^4$
• $\left(5^2\right)^3$

Answer Key

• $\left(3^4\right)^2 = 3^{4 \cdot 2} = 3^8$
• $\left(2^3\right)^4 = 2^{3 \cdot 4} = 2^{12}$
• $\left(5^2\right)^3 = 5^{2 \cdot 3} = 5^6$

Q: What is the power rule in mathematics?

A: The power rule is a fundamental concept in mathematics that states that when raising a power to another power, we multiply the exponents. In mathematical notation, this can be represented as:

$(a^m)^n = a^{m \cdot n}$

Q: How do I apply the power rule to simplify an expression?

A: To apply the power rule, simply multiply the exponents of the base number. For example, to simplify the expression $\left(4^2\right)^5$, we would multiply the exponents $2$ and $5$ to get $10$.

$\left(4^2\right)^5 = 4^{2 \cdot 5} = 4^{10}$

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

A: The power rule and the product rule are two separate concepts in mathematics. The power rule states that when raising a power to another power, we multiply the exponents, while the product rule states that when multiplying two powers with the same base, we add the exponents.

For example, to simplify the expression $2^3 \cdot 2^4$, we would use the product rule to add the exponents:

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

Q: Can I use the power rule to simplify expressions with negative exponents?

A: Yes, you can use the power rule to simplify expressions with negative exponents. When raising a power to another power, the negative exponent is multiplied by the power.

For example, to simplify the expression $\left(4^{-2}\right)^5$, we would multiply the exponents $-2$ and $5$ to get $-10$.

$\left(4^{-2}\right)^5 = 4^{-2 \cdot 5} = 4^{-10}$

Q: How do I evaluate an expression with a power of 0?

A: When evaluating an expression with a power of 0, the result is always 1. This is because any number raised to the power of 0 is equal to 1.

For example, to evaluate the expression $4^0$, we would get:

$4^0 = 1$

Q: Can I use the power rule to simplify expressions with fractional exponents?

A: Yes, you can use the power rule to simplify expressions with fractional exponents. When raising a power to another power, the fractional exponent is multiplied by the power.

For example, to simplify the expression $\left(4^{1/2}\right)^5$, we would multiply the exponents $1/2$ and $5$ to get $5/2$.

$\left(4^{1/2}\right)^5 = 4^{(1/2) \cdot 5} = 4^{5/2}$

Q: What are some common mistakes to avoid when applying the power rule?

A: Some common mistakes to avoid when applying the power rule include:

• Multiplying the exponents instead of adding them (e.g., $\left(4^2\right)^5 = 4^{2+5}$ instead of $4^{2 \cdot 5}$)
• Forgetting to multiply the exponents when raising a power to another power
• Not simplifying the expression after applying the power rule

By following the steps outlined in this article and practicing with the provided examples, you will become proficient in applying the power rule to simplify complex exponential expressions.