Juanita Simplified The Given Expression Using The Following Steps:$\[ \begin{aligned} \left(4 E^{3 X}\right)^2 & = 4 E^{(3 X)(2)} \\ & = 4 E^{6 X} \end{aligned} \\]What Did Juanita Do Wrong, And How Could She Fix Her Error?A. She Should Have

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Understanding Exponent Rules

When working with exponents, it's essential to apply the rules correctly to avoid errors. In the given expression, Juanita attempted to simplify the exponent using the power rule. However, she made a mistake that can be easily corrected.

The Power Rule: A Review

The power rule states that for any numbers a and b and any variable x, the following rule applies:

$(a^b)^c = a^{bc}$

This rule allows us to simplify expressions with multiple exponents.

Juanita's Error

Let's examine Juanita's work:

{ \begin{aligned} \left(4 e^{3 x}\right)^2 & = 4 e^{(3 x)(2)} \\ & = 4 e^{6 x} \end{aligned} \}

Juanita applied the power rule incorrectly. She multiplied the exponent 3x by 2, resulting in 6x. However, this is not the correct application of the power rule.

Correcting the Error

To simplify the expression correctly, Juanita should have applied the power rule as follows:

{ \begin{aligned} \left(4 e^{3 x}\right)^2 & = (4)^2 (e^{3 x})^2 \\ & = 16 e^{2(3 x)} \\ & = 16 e^{6 x} \end{aligned} \}

In this corrected version, Juanita applied the power rule to the exponent 3x, multiplying it by 2 to get 6x. She also squared the coefficient 4 to get 16.

Conclusion

Juanita's mistake was a simple error in applying the power rule. By following the correct steps, she can simplify the expression correctly. Remember to always apply the power rule carefully, making sure to multiply the exponents and square the coefficients.

Common Mistakes to Avoid

When working with exponents, it's easy to make mistakes. Here are some common errors to watch out for:

  • Incorrectly applying the power rule: Make sure to multiply the exponents and square the coefficients.
  • Forgetting to square the coefficient: Don't forget to square the coefficient when applying the power rule.
  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions with exponents.

Practice Makes Perfect

Simplifying expressions with exponents requires practice. Try working through some examples to get a feel for the power rule and how to apply it correctly.

Example 1

Simplify the expression: (2x3)2(2x^3)^2

Solution

Using the power rule, we get:

(2x3)2=(2)2(x3)2=4x6(2x^3)^2 = (2)^2 (x^3)^2 = 4x^6

Example 2

Simplify the expression: (3e2x)2(3e^2x)^2

Solution

Using the power rule, we get:

(3e2x)2=(3)2(e2x)2=9e4x2(3e^2x)^2 = (3)^2 (e^2x)^2 = 9e^4x^2

Understanding Exponent Rules

When working with exponents, it's essential to apply the rules correctly to avoid errors. In the given expression, Juanita attempted to simplify the exponent using the power rule. However, she made a mistake that can be easily corrected.

The Power Rule: A Review

The power rule states that for any numbers a and b and any variable x, the following rule applies:

$(a^b)^c = a^{bc}$

This rule allows us to simplify expressions with multiple exponents.

Juanita's Error

Let's examine Juanita's work:

{ \begin{aligned} \left(4 e^{3 x}\right)^2 & = 4 e^{(3 x)(2)} \\ & = 4 e^{6 x} \end{aligned} \}

Juanita applied the power rule incorrectly. She multiplied the exponent 3x by 2, resulting in 6x. However, this is not the correct application of the power rule.

Correcting the Error

To simplify the expression correctly, Juanita should have applied the power rule as follows:

{ \begin{aligned} \left(4 e^{3 x}\right)^2 & = (4)^2 (e^{3 x})^2 \\ & = 16 e^{2(3 x)} \\ & = 16 e^{6 x} \end{aligned} \}

In this corrected version, Juanita applied the power rule to the exponent 3x, multiplying it by 2 to get 6x. She also squared the coefficient 4 to get 16.

Q&A: Simplifying Exponents

Q: What is the power rule in exponents?

A: The power rule states that for any numbers a and b and any variable x, the following rule applies: $(a^b)^c = a^{bc}$

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

A: To apply the power rule, multiply the exponents and square the coefficients. For example, if you have the expression (2x^3)^2, you would multiply the exponent 3 by 2 to get 6, and square the coefficient 2 to get 4.

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

A: The power rule applies to expressions with multiple exponents, while the product rule applies to expressions with multiple factors. For example, if you have the expression (2x^3)(4x^2), you would apply the product rule to get 8x^5.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by applying the power rule. For example, if you have the expression (2x^(-3))^2, you would multiply the exponent -3 by 2 to get -6, and square the coefficient 2 to get 4.

Q: How do I simplify an expression with a variable in the exponent?

A: To simplify an expression with a variable in the exponent, you can apply the power rule by multiplying the exponents. For example, if you have the expression (2x^3)^2, you would multiply the exponent 3 by 2 to get 6, and square the coefficient 2 to get 4.

Q: Can I simplify an expression with a coefficient in the exponent?

A: Yes, you can simplify an expression with a coefficient in the exponent by applying the power rule. For example, if you have the expression (2x^3)^2, you would multiply the exponent 3 by 2 to get 6, and square the coefficient 2 to get 4.

Q: How do I know when to apply the power rule?

A: You should apply the power rule when you have an expression with multiple exponents. For example, if you have the expression (2x^3)^2, you would apply the power rule to simplify the expression.

Practice Makes Perfect

Simplifying expressions with exponents requires practice. Try working through some examples to get a feel for the power rule and how to apply it correctly.

Example 1

Simplify the expression: (2x3)2(2x^3)^2

Solution

Using the power rule, we get:

(2x3)2=(2)2(x3)2=4x6(2x^3)^2 = (2)^2 (x^3)^2 = 4x^6

Example 2

Simplify the expression: (3e2x)2(3e^2x)^2

Solution

Using the power rule, we get:

(3e2x)2=(3)2(e2x)2=9e4x2(3e^2x)^2 = (3)^2 (e^2x)^2 = 9e^4x^2

By following the correct steps and practicing regularly, you'll become more comfortable simplifying expressions with exponents.