Simplify The Expression And Answer With The Correct Letter:$\left(\frac{4 T^5 U^3}{2 T^6}\right)^3$A. $\frac{2 U^9}{t^3}$B. $8 T^3 U^9$C. $\frac{8 U^6}{t}$D. $\frac{8 U^9}{t^3}$E. None Of These

Feb 26, 2025 by ADMIN 194 views

Understanding the Problem

The given problem involves simplifying an algebraic expression and selecting the correct answer from the options provided. The expression to be simplified is $\left(\frac{4 t^5 u^3}{2 t^6}\right)^3$ . To simplify this expression, we need to apply the rules of exponents and perform the necessary calculations.

Step 1: Simplify the Expression Inside the Parentheses

The first step is to simplify the expression inside the parentheses. We can start by canceling out the common factors in the numerator and denominator.

$\left(\frac{4 t^5 u^3}{2 t^6}\right)^3$

We can rewrite the expression as:

$\left(\frac{2 \cdot 2 \cdot t^5 \cdot u^3}{2 \cdot t^6}\right)^3$

Now, we can cancel out the common factor of 2 in the numerator and denominator:

$\left(\frac{2 \cdot t^5 \cdot u^3}{t^6}\right)^3$

Next, we can simplify the expression by canceling out the common factor of $t^5$ in the numerator and denominator:

$\left(\frac{2 \cdot u^3}{t}\right)^3$

Step 2: Apply the Power Rule of Exponents

The next step is to apply the power rule of exponents, which states that $(a^m)^n = a^{m \cdot n}$ . We can apply this rule to the expression:

$\left(\frac{2 \cdot u^3}{t}\right)^3$

Using the power rule, we get:

$\frac{2^3 \cdot (u^3)^3}{t^3}$

Now, we can simplify the expression by evaluating the exponent:

$\frac{8 \cdot u^9}{t^3}$

Step 3: Select the Correct Answer

The final step is to select the correct answer from the options provided. Based on our calculations, we can see that the simplified expression is $\frac{8 \cdot u^9}{t^3}$ . Therefore, the correct answer is:

D. $\frac{8 u^9}{t^3}$

Conclusion

In this article, we simplified the expression $\left(\frac{4 t^5 u^3}{2 t^6}\right)^3$ and selected the correct answer from the options provided. We applied the rules of exponents and performed the necessary calculations to simplify the expression. The final answer is $\frac{8 u^9}{t^3}$ .

Key Takeaways

To simplify an algebraic expression, we need to apply the rules of exponents and perform the necessary calculations.
The power rule of exponents states that $(a^m)^n = a^{m \cdot n}$ .
We can simplify an expression by canceling out common factors and applying the power rule of exponents.

Frequently Asked Questions

What is the power rule of exponents?
How do we simplify an algebraic expression?
What is the final answer to the given problem?

Answers

The power rule of exponents states that $(a^m)^n = a^{m \cdot n}$ .
To simplify an algebraic expression, we need to apply the rules of exponents and perform the necessary calculations.
The final answer to the given problem is $\frac{8 u^9}{t^3}$ .
Simplify the Expression and Answer with the Correct Letter: Q&A ================================================================

Understanding the Problem

Q&A Session

Q: What is the power rule of exponents?

A: The power rule of exponents states that $(a^m)^n = a^{m \cdot n}$ . This means that when we raise a power to another power, we multiply the exponents.

Q: How do we simplify an algebraic expression?

A: To simplify an algebraic expression, we need to apply the rules of exponents and perform the necessary calculations. We can start by canceling out common factors in the numerator and denominator, and then apply the power rule of exponents.

Q: What is the final answer to the given problem?

A: The final answer to the given problem is $\frac{8 u^9}{t^3}$ .

Q: Can you explain the steps to simplify the expression?

A: Yes, certainly. Here are the steps to simplify the expression:

Simplify the expression inside the parentheses by canceling out common factors in the numerator and denominator.
Apply the power rule of exponents to simplify the expression.
Evaluate the exponent to get the final answer.

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

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

Not canceling out common factors in the numerator and denominator.
Not applying the power rule of exponents correctly.
Not evaluating the exponent correctly.

Q: How can we use the power rule of exponents in real-life situations?

A: The power rule of exponents can be used in a variety of real-life situations, such as:

Calculating the area of a circle: $A = \pi r^2$
Calculating the volume of a sphere: $V = \frac{4}{3} \pi r^3$
Calculating the surface area of a cube: $A = 6s^2$

Q: What are some tips for simplifying expressions?

A: Some tips for simplifying expressions include:

Start by simplifying the expression inside the parentheses.
Apply the power rule of exponents to simplify the expression.
Evaluate the exponent to get the final answer.
Check your work to make sure you have simplified the expression correctly.

Conclusion

In this article, we simplified the expression $\left(\frac{4 t^5 u^3}{2 t^6}\right)^3$ and selected the correct answer from the options provided. We also answered some frequently asked questions about simplifying expressions and the power rule of exponents. We hope this article has been helpful in understanding the concept of simplifying expressions and the power rule of exponents.

Key Takeaways

The power rule of exponents states that $(a^m)^n = a^{m \cdot n}$ .
To simplify an algebraic expression, we need to apply the rules of exponents and perform the necessary calculations.
The final answer to the given problem is $\frac{8 u^9}{t^3}$ .

Frequently Asked Questions

What is the power rule of exponents?
How do we simplify an algebraic expression?
What is the final answer to the given problem?

Answers

The power rule of exponents states that $(a^m)^n = a^{m \cdot n}$ .
To simplify an algebraic expression, we need to apply the rules of exponents and perform the necessary calculations.
The final answer to the given problem is $\frac{8 u^9}{t^3}$ .