Examine Tamika's Work Below. What Is Tamika's Error?${ \begin{aligned} \frac{\operatorname{lia} A^{-9}}{\sqrt{11} B^{-1}} & = \frac{3 A^{-2} B^{-11}}{5} \ & = \frac{3}{5 A^{211}} \end{aligned} }$A. She Added The Exponents.B. She Simplified

Introduction

Tamika's work on simplifying an algebraic expression has been provided, but it appears to contain an error. The given expression involves exponents and fractions, which can be simplified using the rules of exponentiation and fraction manipulation. In this article, we will examine Tamika's work and identify the error she made.

The Given Expression

The given expression is:

{ \begin{aligned} \frac{\operatorname{lia} a^{-9}}{\sqrt{11} b^{-1}} & = \frac{3 a^{-2} b^{-11}}{5} \\ & = \frac{3}{5 a^{211}} \end{aligned} \}

Step-by-Step Analysis

To simplify the given expression, we need to follow the order of operations (PEMDAS):

1. Evaluate the expressions inside the parentheses.
2. Simplify the exponents.
3. Simplify the fractions.

Let's analyze each step:

Step 1: Evaluate the Expressions Inside the Parentheses

The expression inside the parentheses is $\operatorname{lia} a^{-9}$. However, there is no definition or explanation for the term $\operatorname{lia}$. Assuming it's a typo or a mistake, we can ignore it and focus on the rest of the expression.

Step 2: Simplify the Exponents

The expression contains two exponents: $a^{-9}$ and $b^{-1}$. When dividing two exponential expressions with the same base, we subtract the exponents:

$a^{-9} \div b^{-1} = a^{-9 - (-1)} = a^{-8}$

Step 3: Simplify the Fractions

The expression contains two fractions: $\frac{a^{-8}}{\sqrt{11} b^{-1}}$ and $\frac{3 a^{-2} b^{-11}}{5}$. To simplify the fractions, we can multiply the numerator and denominator by the reciprocal of the denominator:

$\frac{a^{-8}}{\sqrt{11} b^{-1}} = \frac{a^{-8} \cdot \sqrt{11} b}{\sqrt{11} \cdot \sqrt{11} b^{-1}} = \frac{a^{-8} \cdot \sqrt{11} b}{11 b^{-1}}$

Step 4: Simplify the Exponents Further

Now we can simplify the exponents further:

$a^{-8} \cdot \sqrt{11} b = \frac{\sqrt{11} b}{a^8}$

Step 5: Combine the Fractions

We can now combine the two fractions:

$\frac{\sqrt{11} b}{a^8} \div \frac{3 a^{-2} b^{-11}}{5} = \frac{\sqrt{11} b}{a^8} \cdot \frac{5}{3 a^{-2} b^{-11}}$

Step 6: Simplify the Exponents and Fractions

Now we can simplify the exponents and fractions:

$\frac{\sqrt{11} b}{a^8} \cdot \frac{5}{3 a^{-2} b^{-11}} = \frac{\sqrt{11} b \cdot 5}{a^8 \cdot 3 a^{-2} b^{-11}} = \frac{5 \sqrt{11} b}{3 a^{10} b^{-11}}$

Step 7: Simplify the Final Expression

Finally, we can simplify the final expression:

$\frac{5 \sqrt{11} b}{3 a^{10} b^{-11}} = \frac{5 \sqrt{11}}{3 a^{10} b^{-12}}$

Conclusion

Tamika's error was in simplifying the exponents and fractions. She incorrectly added the exponents and did not follow the order of operations. The correct simplification of the given expression is:

$\frac{5 \sqrt{11}}{3 a^{10} b^{-12}}$

Discussion

Introduction

In our previous article, we examined Tamika's work on simplifying an algebraic expression and identified the error she made. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q: What is the correct simplification of the given expression?

A: The correct simplification of the given expression is:

$\frac{5 \sqrt{11}}{3 a^{10} b^{-12}}$

Q: Why did Tamika incorrectly add the exponents?

A: Tamika incorrectly added the exponents because she did not follow the order of operations (PEMDAS). When dividing two exponential expressions with the same base, we subtract the exponents, not add them.

Q: What is the rule for simplifying exponents?

A: The rule for simplifying exponents is:

• When multiplying two exponential expressions with the same base, we add the exponents.
• When dividing two exponential expressions with the same base, we subtract the exponents.

Q: How do we simplify fractions with exponents?

A: To simplify fractions with exponents, we can multiply the numerator and denominator by the reciprocal of the denominator. This will allow us to simplify the exponents and fractions.

Q: What is the correct order of operations for simplifying algebraic expressions?

A: The correct order of operations for simplifying algebraic expressions is:

1. Evaluate the expressions inside the parentheses.
2. Simplify the exponents.
3. Simplify the fractions.

Q: Why is it important to follow the order of operations?

A: It is important to follow the order of operations because it ensures that we simplify the expression correctly and avoid errors. If we do not follow the order of operations, we may end up with an incorrect simplification.

Q: Can you provide an example of a similar problem?

A: Here is an example of a similar problem:

$\frac{\operatorname{lia} a^{-5}}{\sqrt{11} b^{-2}} = \frac{2 a^{-3} b^{-7}}{4}$

Can you simplify this expression and identify any errors?

Answer

To simplify this expression, we need to follow the order of operations:

1. Evaluate the expressions inside the parentheses.
2. Simplify the exponents.
3. Simplify the fractions.

Using the rules of exponentiation and fraction manipulation, we can simplify the expression as follows:

$\frac{\operatorname{lia} a^{-5}}{\sqrt{11} b^{-2}} = \frac{2 a^{-3} b^{-7}}{4} = \frac{2}{4 a^3 b^7}$

However, there is an error in the original expression. The term $\operatorname{lia}$ is not defined, and the expression contains a typo. The correct simplification of the expression is:

$\frac{2}{4 a^3 b^7} = \frac{1}{2 a^3 b^7}$

Conclusion

In this Q&A article, we provided additional information and clarification on the topic of simplifying algebraic expressions. We also provided an example of a similar problem and walked through the solution step-by-step. We hope this article has been helpful in understanding the rules of exponentiation and fraction manipulation.