Simplify The Expression:${ \left(\frac{5 X^3}{2 X^6 Y 4}\right) 2 }$Write Your Answer Using Only Positive Exponents.

$$

Understanding the Problem

When simplifying the given expression, we need to apply the rules of exponents and fractions. The expression involves a fraction raised to the power of 2, which means we need to apply the power rule for fractions and exponents.

Applying the Power Rule for Fractions

The power rule for fractions states that when a fraction is raised to a power, we can raise the numerator and denominator to that power separately. In this case, we have:

$\left(\frac{5 x^3}{2 x^6 y^4}\right)^2$

Using the power rule, we can rewrite this expression as:

$\frac{(5 x^3)^2}{(2 x^6 y^4)^2}$

Simplifying the Numerator and Denominator

Now, let's simplify the numerator and denominator separately. We can use the power rule for exponents, which states that when an exponent is raised to a power, we can multiply the exponents. In this case, we have:

$(5 x^3)^2 = 5^2 (x^3)^2$

Using the power rule for exponents, we can rewrite this expression as:

$25 x^6$

Similarly, we can simplify the denominator as:

$(2 x^6 y^4)^2 = 2^2 (x^6)^2 (y^4)^2$

Using the power rule for exponents, we can rewrite this expression as:

$4 x^{12} y^8$

Combining the Simplified Numerator and Denominator

Now, let's combine the simplified numerator and denominator to get the final expression:

$\frac{25 x^6}{4 x^{12} y^8}$

Simplifying the Expression Further

We can simplify the expression further by applying the rule for dividing like bases with exponents. When we divide like bases with exponents, we can subtract the exponents. In this case, we have:

$\frac{25 x^6}{4 x^{12} y^8}$

Using the rule for dividing like bases with exponents, we can rewrite this expression as:

$\frac{25}{4 x^6 y^8}$

Writing the Answer Using Only Positive Exponents

Finally, let's rewrite the answer using only positive exponents. We can do this by applying the rule for moving exponents from the denominator to the numerator. When we move an exponent from the denominator to the numerator, we can change the sign of the exponent. In this case, we have:

$\frac{25}{4 x^6 y^8}$

Using the rule for moving exponents from the denominator to the numerator, we can rewrite this expression as:

$\frac{25 y^8}{4 x^6}$

Conclusion

In conclusion, we have simplified the given expression using the rules of exponents and fractions. We applied the power rule for fractions and exponents, simplified the numerator and denominator, and combined the simplified expressions to get the final answer. We also rewrote the answer using only positive exponents.

Final Answer

The final answer is $\boxed{\frac{25 y^8}{4 x^6}}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Apply the power rule for fractions: $\left(\frac{5 x^3}{2 x^6 y^4}\right)^2 = \frac{(5 x^3)^2}{(2 x^6 y^4)^2}$
2. Simplify the numerator and denominator: $(5 x^3)^2 = 25 x^6$ and $(2 x^6 y^4)^2 = 4 x^{12} y^8$
3. Combine the simplified numerator and denominator: $\frac{25 x^6}{4 x^{12} y^8}$
4. Simplify the expression further: $\frac{25}{4 x^6 y^8}$
5. Rewrite the answer using only positive exponents: $\frac{25 y^8}{4 x^6}$

Frequently Asked Questions

• Q: What is the power rule for fractions? A: The power rule for fractions states that when a fraction is raised to a power, we can raise the numerator and denominator to that power separately.
• Q: How do we simplify the numerator and denominator? A: We can simplify the numerator and denominator by applying the power rule for exponents and simplifying the resulting expressions.
• Q: How do we combine the simplified numerator and denominator? A: We can combine the simplified numerator and denominator by dividing the numerator by the denominator.
• Q: How do we rewrite the answer using only positive exponents? A: We can rewrite the answer using only positive exponents by applying the rule for moving exponents from the denominator to the numerator.
  

$$

Understanding the Problem

When simplifying the given expression, we need to apply the rules of exponents and fractions. The expression involves a fraction raised to the power of 2, which means we need to apply the power rule for fractions and exponents.

Q&A

Q: What is the power rule for fractions?

A: The power rule for fractions states that when a fraction is raised to a power, we can raise the numerator and denominator to that power separately.

Q: How do we simplify the numerator and denominator?

A: We can simplify the numerator and denominator by applying the power rule for exponents and simplifying the resulting expressions.

Q: What is the rule for dividing like bases with exponents?

A: When we divide like bases with exponents, we can subtract the exponents.

Q: How do we rewrite the answer using only positive exponents?

A: We can rewrite the answer using only positive exponents by applying the rule for moving exponents from the denominator to the numerator.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{\frac{25 y^8}{4 x^6}}$.

Q: Can you provide a step-by-step solution to the problem?

A: Here is the step-by-step solution to the problem:

1. Apply the power rule for fractions: $\left(\frac{5 x^3}{2 x^6 y^4}\right)^2 = \frac{(5 x^3)^2}{(2 x^6 y^4)^2}$
2. Simplify the numerator and denominator: $(5 x^3)^2 = 25 x^6$ and $(2 x^6 y^4)^2 = 4 x^{12} y^8$
3. Combine the simplified numerator and denominator: $\frac{25 x^6}{4 x^{12} y^8}$
4. Simplify the expression further: $\frac{25}{4 x^6 y^8}$
5. Rewrite the answer using only positive exponents: $\frac{25 y^8}{4 x^6}$

Q: What are some common mistakes to avoid when simplifying expressions with exponents?

A: Some common mistakes to avoid when simplifying expressions with exponents include:

• Not applying the power rule for fractions correctly
• Not simplifying the numerator and denominator correctly
• Not combining the simplified numerator and denominator correctly
• Not rewriting the answer using only positive exponents correctly

Q: How can I practice simplifying expressions with exponents?

A: You can practice simplifying expressions with exponents by working through example problems and exercises. You can also try simplifying expressions with exponents on your own and then checking your work with a calculator or by asking a teacher or tutor for help.

Conclusion

In conclusion, simplifying expressions with exponents requires applying the power rule for fractions and exponents, simplifying the numerator and denominator, and combining the simplified expressions to get the final answer. By following the steps outlined in this article and practicing simplifying expressions with exponents, you can become more confident and proficient in simplifying expressions with exponents.

Final Answer

The final answer is $\boxed{\frac{25 y^8}{4 x^6}}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Apply the power rule for fractions: $\left(\frac{5 x^3}{2 x^6 y^4}\right)^2 = \frac{(5 x^3)^2}{(2 x^6 y^4)^2}$
2. Simplify the numerator and denominator: $(5 x^3)^2 = 25 x^6$ and $(2 x^6 y^4)^2 = 4 x^{12} y^8$
3. Combine the simplified numerator and denominator: $\frac{25 x^6}{4 x^{12} y^8}$
4. Simplify the expression further: $\frac{25}{4 x^6 y^8}$
5. Rewrite the answer using only positive exponents: $\frac{25 y^8}{4 x^6}$