Simplify The Expression:${ 8^{-\frac{2}{3}} }$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common ways to simplify expressions is by using exponent rules. In this article, we will focus on simplifying the expression $8^{-\frac{2}{3}}$ using exponent rules and properties.

Understanding Exponents

Before we dive into simplifying the expression, let's quickly review what exponents are. An exponent is a small number that is written above and to the right of a larger number, indicating how many times the larger number should be multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$. In this case, the exponent $3$ tells us how many times to multiply $2$ by itself.

Simplifying Negative Exponents

Now, let's talk about negative exponents. A negative exponent is a way of expressing a fraction with a power of a number. For example, $2^{-3}$ can be written as $\frac{1}{2^3}$, which equals $\frac{1}{8}$. In general, a negative exponent can be rewritten as a fraction with the power of the number in the denominator.

Applying Exponent Rules to Simplify the Expression

Now that we have a good understanding of exponents and negative exponents, let's apply these rules to simplify the expression $8^{-\frac{2}{3}}$. To simplify this expression, we need to use the rule that states $a^{-n} = \frac{1}{a^n}$.

Step 1: Rewrite the Expression with a Negative Exponent

Using the rule mentioned above, we can rewrite the expression $8^{-\frac{2}{3}}$ as $\frac{1}{8^{\frac{2}{3}}}$.

Step 2: Simplify the Expression Inside the Parentheses

Now, let's simplify the expression inside the parentheses, $8^{\frac{2}{3}}$. To do this, we need to use the rule that states $(a^m)^n = a^{m \cdot n}$. In this case, we have $(8^{\frac{1}{3}})^2$, which equals $8^{\frac{2}{3}}$.

Step 3: Simplify the Expression Further

Now that we have simplified the expression inside the parentheses, we can simplify the entire expression. We have $\frac{1}{8^{\frac{2}{3}}} = \frac{1}{(8^{\frac{1}{3}})^2}$. Using the rule mentioned above, we can rewrite this as $\frac{1}{(2^3)^{\frac{2}{3}}}$.

Step 4: Simplify the Expression Inside the Parentheses Again

Now, let's simplify the expression inside the parentheses again, $(2^3)^{\frac{2}{3}}$. Using the rule mentioned above, we can rewrite this as $2^{3 \cdot \frac{2}{3}}$, which equals $2^2$.

Step 5: Simplify the Expression Further

Now that we have simplified the expression inside the parentheses again, we can simplify the entire expression. We have $\frac{1}{(2^3)^{\frac{2}{3}}} = \frac{1}{2^2}$.

Step 6: Simplify the Expression to Its Final Form

Finally, let's simplify the expression to its final form. We have $\frac{1}{2^2} = \frac{1}{4}$.

Conclusion

In this article, we simplified the expression $8^{-\frac{2}{3}}$ using exponent rules and properties. We started by rewriting the expression with a negative exponent, then simplified the expression inside the parentheses, and finally simplified the entire expression to its final form. The final answer is $\frac{1}{4}$.

Frequently Asked Questions

• What is the rule for simplifying negative exponents? A negative exponent can be rewritten as a fraction with the power of the number in the denominator.
• How do you simplify an expression with a negative exponent? To simplify an expression with a negative exponent, you need to use the rule that states $a^{-n} = \frac{1}{a^n}$.
• What is the final answer to the expression $8^{-\frac{2}{3}}$? The final answer is $\frac{1}{4}$.

Additional Resources

• Exponent Rules: A Comprehensive Guide
• Simplifying Expressions with Negative Exponents
• Math Problems with Exponents and Fractions
  

Introduction

In our previous article, we simplified the expression $8^{-\frac{2}{3}}$ using exponent rules and properties. In this article, we will answer some frequently asked questions about simplifying expressions with negative exponents.

Q&A

Q: What is the rule for simplifying negative exponents?

A: A negative exponent can be rewritten as a fraction with the power of the number in the denominator. This rule is expressed as $a^{-n} = \frac{1}{a^n}$.

Q: How do you simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you need to use the rule that states $a^{-n} = \frac{1}{a^n}$. This means that you can rewrite the expression with a negative exponent as a fraction with the power of the number in the denominator.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the number is being multiplied by itself a certain number of times, while a negative exponent indicates that the number is being divided by itself a certain number of times.

Q: Can you give an example of simplifying an expression with a negative exponent?

A: Yes, let's consider the expression $2^{-3}$. Using the rule mentioned above, we can rewrite this expression as $\frac{1}{2^3}$. Simplifying further, we get $\frac{1}{8}$.

Q: How do you simplify an expression with a negative exponent and a fraction?

A: To simplify an expression with a negative exponent and a fraction, you need to follow the order of operations (PEMDAS). First, simplify the fraction, then apply the exponent rule.

Q: Can you give an example of simplifying an expression with a negative exponent and a fraction?

A: Yes, let's consider the expression $\frac{1}{2^{-3}}$. Using the rule mentioned above, we can rewrite this expression as $2^3$. Simplifying further, we get $8$.

Q: What is the final answer to the expression $8^{-\frac{2}{3}}$?

A: The final answer is $\frac{1}{4}$.

Q: How do you simplify an expression with a negative exponent and a variable?

A: To simplify an expression with a negative exponent and a variable, you need to follow the same rules as before. First, rewrite the expression with a negative exponent as a fraction with the power of the variable in the denominator.

Q: Can you give an example of simplifying an expression with a negative exponent and a variable?

A: Yes, let's consider the expression $x^{-2}$. Using the rule mentioned above, we can rewrite this expression as $\frac{1}{x^2}$.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with negative exponents. We covered topics such as the rule for simplifying negative exponents, how to simplify expressions with negative exponents, and how to simplify expressions with negative exponents and fractions or variables.

Frequently Asked Questions

• What is the rule for simplifying negative exponents?
• How do you simplify an expression with a negative exponent?
• What is the difference between a positive exponent and a negative exponent?
• Can you give an example of simplifying an expression with a negative exponent?
• How do you simplify an expression with a negative exponent and a fraction?
• Can you give an example of simplifying an expression with a negative exponent and a fraction?
• What is the final answer to the expression $8^{-\frac{2}{3}}$?
• How do you simplify an expression with a negative exponent and a variable?
• Can you give an example of simplifying an expression with a negative exponent and a variable?

Additional Resources

• Exponent Rules: A Comprehensive Guide
• Simplifying Expressions with Negative Exponents
• Math Problems with Exponents and Fractions