Why Approach A is Incorrect

Oct 13, 2025 by talynt.org 57 views

Understanding the Power Rule

The power rule is a fundamental concept in algebra that allows us to simplify exponential expressions by applying the properties of exponents. In this article, we will explore the power rule and its application to simplify the expression $\left(7^{10}\right)^5$ . We will examine three different approaches to simplify this expression and determine which one applies the power rule properly.

The Power Rule: A Brief Overview

The power rule states that for any non-zero number $a$ and integers $m$ and $n$ , the following equation holds:

\left(a^m\right)^n = a^{m \cdot n}

This rule allows us to simplify exponential expressions by multiplying the exponents when the base is the same.

Simplifying the Expression $\left(7^{10}\right)^5$

Now, let's apply the power rule to simplify the expression $\left(7^{10}\right)^5$ . We will examine three different approaches to simplify this expression.

Approach A: $\left(7^{10}\right)^5=7^{10 \div 5}=7^2$

This approach suggests that we can simplify the expression by dividing the exponent $10$ by $5$ , resulting in $7^2$ . However, this approach is incorrect because it does not apply the power rule properly.

Why Approach A is Incorrect

The power rule states that we should multiply the exponents, not divide them. Therefore, Approach A is incorrect because it does not follow the power rule.

Approach B: $\left(7^{10}\right)^5=7^{10-5}=7^5$

This approach suggests that we can simplify the expression by subtracting the exponents $10$ and $5$ , resulting in $7^5$ . However, this approach is also incorrect because it does not apply the power rule properly.

Why Approach B is Incorrect

The power rule states that we should multiply the exponents, not subtract them. Therefore, Approach B is incorrect because it does not follow the power rule.

Approach C: $\left(7^{10}\right)^5=7^{10 \cdot 5}=7^{50}$

This approach suggests that we can simplify the expression by multiplying the exponents $10$ and $5$ , resulting in $7^{50}$ . This approach is correct because it applies the power rule properly.

Why Approach C is Correct

The power rule states that we should multiply the exponents when the base is the same. Therefore, Approach C is correct because it follows the power rule.

Conclusion

In conclusion, the correct approach to simplify the expression $\left(7^{10}\right)^5$ is to multiply the exponents $10$ and $5$ , resulting in $7^{50}$ . This approach applies the power rule properly and follows the properties of exponents.

Common Mistakes to Avoid

When simplifying exponential expressions using the power rule, it's essential to avoid common mistakes such as dividing or subtracting the exponents. Instead, always multiply the exponents when the base is the same.

Real-World Applications

The power rule has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the power rule is used to calculate the energy of a system, while in engineering, it's used to design electrical circuits.

Practice Problems

To practice simplifying exponential expressions using the power rule, try the following problems:

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

Answer Key

$\left(2^3\right)^4 = 2^{3 \cdot 4} = 2^{12}$
$\left(5^2\right)^3 = 5^{2 \cdot 3} = 5^6$
$\left(3^5\right)^2 = 3^{5 \cdot 2} = 3^{10}$

Q: What is the power rule in algebra?

A: The power rule is a fundamental concept in algebra that allows us to simplify exponential expressions by applying the properties of exponents. It states that for any non-zero number $a$ and integers $m$ and $n$ , the following equation holds:

\left(a^m\right)^n = a^{m \cdot n}

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

A: To apply the power rule, simply multiply the exponents when the base is the same. For example, to simplify the expression $\left(7^{10}\right)^5$ , you would multiply the exponents $10$ and $5$ , resulting in $7^{50}$ .

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:

Dividing the exponents instead of multiplying them
Subtracting the exponents instead of multiplying them
Not following the order of operations (PEMDAS)

Q: How do I know if I'm applying the power rule correctly?

A: To ensure you're applying the power rule correctly, follow these steps:

Identify the base and exponents in the expression
Multiply the exponents when the base is the same
Simplify the resulting expression

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

A: Yes, you can apply the power rule to expressions with negative exponents. For example, to simplify the expression $\left(2^{-3}\right)^4$ , you would multiply the exponents $-3$ and $4$ , resulting in $2^{-12}$ .

Q: How do I apply the power rule to expressions with fractional exponents?

A: To apply the power rule to expressions with fractional exponents, simply multiply the exponents when the base is the same. For example, to simplify the expression $\left(3^{1/2}\right)^3$ , you would multiply the exponents $1/2$ and $3$ , resulting in $3^{3/2}$ .

Q: Can I apply the power rule to expressions with variables as exponents?

A: Yes, you can apply the power rule to expressions with variables as exponents. For example, to simplify the expression $\left(x^2\right)^3$ , you would multiply the exponents $2$ and $3$ , resulting in $x^6$ .

Q: How do I apply the power rule to expressions with multiple bases?

A: To apply the power rule to expressions with multiple bases, you must first simplify the expression by applying the power rule to each base separately. For example, to simplify the expression $\left(2^3\right)\left(3^2\right)^4$ , you would first simplify the expression $\left(3^2\right)^4$ by multiplying the exponents $2$ and $4$ , resulting in $3^8$ . Then, you would multiply the exponents $3$ and $8$ , resulting in $2^3 \cdot 3^8$ .

Q: Can I apply the power rule to expressions with radicals?

A: Yes, you can apply the power rule to expressions with radicals. For example, to simplify the expression $\left(\sqrt{2}\right)^4$ , you would multiply the exponents $1/2$ and $4$ , resulting in $\sqrt{2^4} = 4\sqrt{2}$ .

Q: How do I apply the power rule to expressions with absolute values?

A: To apply the power rule to expressions with absolute values, you must first simplify the expression by applying the power rule to the base separately. For example, to simplify the expression $\left|2^3\right|^4$ , you would first simplify the expression $2^3$ by multiplying the exponents $3$ and $4$ , resulting in $2^3 \cdot 2^4 = 2^7$ . Then, you would take the absolute value of the result, resulting in $\left|2^7\right| = 2^7$ .

Q: Can I apply the power rule to expressions with complex numbers?

A: Yes, you can apply the power rule to expressions with complex numbers. For example, to simplify the expression $\left(2+3i\right)^4$ , you would multiply the exponents $1$ and $4$ , resulting in $\left(2+3i\right)^4 = 2^4 + 4 \cdot 2^3 \cdot 3i + 6 \cdot 2^2 \cdot 3^2 i^2 + 4 \cdot 2 \cdot 3^3 i^3 + 3^4 i^4$ .

By following these steps and avoiding common mistakes, you can master the power rule and simplify exponential expressions with ease.