Simplify Fully The Expression Below And Write Your Answer With Positive Indices: ( 125 A 3 E 3 3 ) − 2 \left(\sqrt[3]{125 A^3 E^3}\right)^{-2} ( 3 125 A 3 E 3 ​ ) − 2

Understanding the Problem

When dealing with expressions involving exponents and roots, it's essential to simplify them to their most basic form. In this case, we're given the expression $\left(\sqrt[3]{125 A^3 E^3}\right)^{-2}$, and we need to simplify it and write the answer with positive indices.

Breaking Down the Expression

To simplify the given expression, we need to start by understanding the properties of exponents and roots. The expression involves a cube root, which can be rewritten as an exponent with a fractional index. Specifically, $\sqrt[3]{x} = x^{\frac{1}{3}}$. Applying this property to the given expression, we get:

$\left(\sqrt[3]{125 A^3 E^3}\right)^{-2} = \left((125 A^3 E^3)^{\frac{1}{3}}\right)^{-2}$

Simplifying the Expression

Now that we've rewritten the cube root as an exponent with a fractional index, we can simplify the expression further. When we raise a power to a power, we multiply the exponents. In this case, we have:

$\left((125 A^3 E^3)^{\frac{1}{3}}\right)^{-2} = (125 A^3 E^3)^{\frac{1}{3} \cdot -2}$

Using the property of multiplying exponents, we can simplify the expression to:

$(125 A^3 E^3)^{-\frac{2}{3}}$

Applying the Power of a Product Rule

The expression $(125 A^3 E^3)^{-\frac{2}{3}}$ involves a product of terms raised to a power. To simplify this expression, we can apply the power of a product rule, which states that $(ab)^n = a^n b^n$. In this case, we have:

$(125 A^3 E^3)^{-\frac{2}{3}} = 125^{-\frac{2}{3}} A^{-2} E^{-2}$

Simplifying the Terms

Now that we've applied the power of a product rule, we can simplify the terms further. The term $125^{-\frac{2}{3}}$ can be rewritten as:

$125^{-\frac{2}{3}} = (5^3)^{-\frac{2}{3}} = 5^{-2}$

Using the property of negative exponents, we can rewrite the term $5^{-2}$ as:

$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

Writing the Answer with Positive Indices

Now that we've simplified the expression, we can write the answer with positive indices. The simplified expression is:

$\frac{1}{25} A^{-2} E^{-2}$

To write the answer with positive indices, we can rewrite the negative exponents as positive exponents by taking the reciprocal of the base. Specifically, we have:

$\frac{1}{25} A^{-2} E^{-2} = \frac{1}{A^2 E^2}$

Conclusion

In conclusion, we've simplified the given expression $\left(\sqrt[3]{125 A^3 E^3}\right)^{-2}$ and written the answer with positive indices. The simplified expression is $\frac{1}{A^2 E^2}$.

Final Answer

The final answer is $\boxed{\frac{1}{A^2 E^2}}$.

Discussion

The expression $\left(\sqrt[3]{125 A^3 E^3}\right)^{-2}$ involves a cube root and a negative exponent. To simplify the expression, we need to apply the properties of exponents and roots. Specifically, we need to rewrite the cube root as an exponent with a fractional index and apply the power of a product rule. By simplifying the expression step by step, we can arrive at the final answer of $\frac{1}{A^2 E^2}$.

Related Topics

• Exponents and Roots
• Power of a Product Rule
• Negative Exponents
• Simplifying Expressions

References

• [1] Algebra, 2nd Edition, Michael Artin
• [2] Calculus, 3rd Edition, Michael Spivak
• [3] Mathematics for Computer Science, Eric Lehman, F Thomson Leighton, and Albert R Meyer
  

Understanding the Problem

When dealing with expressions involving exponents and roots, it's essential to simplify them to their most basic form. In this case, we're given the expression $\left(\sqrt[3]{125 A^3 E^3}\right)^{-2}$, and we need to simplify it and write the answer with positive indices.

Q&A

Q: What is the first step in simplifying the given expression?

A: The first step in simplifying the given expression is to rewrite the cube root as an exponent with a fractional index. Specifically, $\sqrt[3]{x} = x^{\frac{1}{3}}$. Applying this property to the given expression, we get:

$\left(\sqrt[3]{125 A^3 E^3}\right)^{-2} = \left((125 A^3 E^3)^{\frac{1}{3}}\right)^{-2}$

Q: How do we simplify the expression further?

A: To simplify the expression further, we can apply the power of a product rule, which states that $(ab)^n = a^n b^n$. In this case, we have:

$\left((125 A^3 E^3)^{\frac{1}{3}}\right)^{-2} = (125 A^3 E^3)^{\frac{1}{3} \cdot -2}$

Using the property of multiplying exponents, we can simplify the expression to:

$(125 A^3 E^3)^{-\frac{2}{3}}$

Q: How do we apply the power of a product rule?

A: To apply the power of a product rule, we need to multiply the exponents of each term. In this case, we have:

$(125 A^3 E^3)^{-\frac{2}{3}} = 125^{-\frac{2}{3}} A^{-2} E^{-2}$

Q: How do we simplify the term $125^{-\frac{2}{3}}$?

A: To simplify the term $125^{-\frac{2}{3}}$, we can rewrite it as:

$125^{-\frac{2}{3}} = (5^3)^{-\frac{2}{3}} = 5^{-2}$

Using the property of negative exponents, we can rewrite the term $5^{-2}$ as:

$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

Q: How do we write the answer with positive indices?

A: To write the answer with positive indices, we can rewrite the negative exponents as positive exponents by taking the reciprocal of the base. Specifically, we have:

$\frac{1}{25} A^{-2} E^{-2} = \frac{1}{A^2 E^2}$

Q: What is the final answer?

A: The final answer is $\boxed{\frac{1}{A^2 E^2}}$.

Discussion

The expression $\left(\sqrt[3]{125 A^3 E^3}\right)^{-2}$ involves a cube root and a negative exponent. To simplify the expression, we need to apply the properties of exponents and roots. Specifically, we need to rewrite the cube root as an exponent with a fractional index and apply the power of a product rule. By simplifying the expression step by step, we can arrive at the final answer of $\frac{1}{A^2 E^2}$.

Related Topics

• Exponents and Roots
• Power of a Product Rule
• Negative Exponents
• Simplifying Expressions

References

• [1] Algebra, 2nd Edition, Michael Artin
• [2] Calculus, 3rd Edition, Michael Spivak
• [3] Mathematics for Computer Science, Eric Lehman, F Thomson Leighton, and Albert R Meyer

Additional Resources

• Khan Academy: Exponents and Roots
• MIT OpenCourseWare: Algebra
• Wolfram Alpha: Exponents and Roots