What Is The Simplest Form Of $\sqrt{a^7}$?A. $a^2 \sqrt{a}$ B. $ A 3 A A^3 \sqrt{a} A 3 A [/tex] C. $a^3 \sqrt{a^2}$ D. $3a \sqrt{a}$
Introduction
When dealing with square roots, it's essential to simplify expressions to their most basic form. This involves breaking down the expression into its prime factors and then simplifying the square root. In this article, we will explore the simplest form of $\sqrt{a^7}$ and examine the different options provided.
Understanding Square Roots
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. When dealing with square roots, it's essential to understand the properties of square roots, including the fact that the square root of a number can be expressed as a product of the square root of a factor and the square root of the remaining factor.
Simplifying $\sqrt{a^7}$
To simplify $\sqrt{a^7}$, we need to break down the expression into its prime factors. We can start by expressing $a^7$ as $a^6 \cdot a$. This allows us to rewrite the expression as $\sqrt{a^6 \cdot a}$.
Applying the Properties of Square Roots
Using the properties of square roots, we can express $\sqrt{a^6 \cdot a}$ as $\sqrt{a^6} \cdot \sqrt{a}$. This is because the square root of a product is equal to the product of the square roots.
Simplifying Further
Now that we have expressed $\sqrt{a^6 \cdot a}$ as $\sqrt{a^6} \cdot \sqrt{a}$, we can simplify further. We know that $\sqrt{a^6}$ can be expressed as $a^3$, because $a^3$ multiplied by $a^3$ equals $a^6$. Therefore, we can rewrite the expression as $a^3 \cdot \sqrt{a}$.
Conclusion
In conclusion, the simplest form of $\sqrt{a^7}$ is $a^3 \sqrt{a}$. This is because we can break down the expression into its prime factors, apply the properties of square roots, and simplify further to arrive at this result.
Comparison with Options
Let's compare our result with the options provided:
- A. $a^2 \sqrta}$$.
- B. $a^3 \sqrt{a}$: This is the correct answer, because we have simplified the expression to its most basic form.
- C. $a^3 \sqrta^2}$$.
- D. $3a \sqrt{a}$: This is not the correct answer, because we have not multiplied the expression by 3.
Final Answer
The final answer is B. $a^3 \sqrt{a}$.
Introduction
In our previous article, we explored the simplest form of $\sqrt{a^7}$. We broke down the expression into its prime factors, applied the properties of square roots, and simplified further to arrive at the result. In this article, we will answer some frequently asked questions about simplifying square roots.
Q: What is the simplest form of $\sqrt{a^4}$?
A: The simplest form of $\sqrt{a^4}$ is $a^2$. This is because we can break down the expression into its prime factors, apply the properties of square roots, and simplify further to arrive at this result.
Q: How do I simplify $\sqrt{a^3}$?
A: To simplify $\sqrt{a^3}$, we can break down the expression into its prime factors. We can express $a^3$ as $a^2 \cdot a$. Then, we can apply the properties of square roots to simplify further. The result is $a \sqrt{a}$.
Q: What is the simplest form of $\sqrt{a^2}$?
A: The simplest form of $\sqrt{a^2}$ is $a$. This is because we can break down the expression into its prime factors, apply the properties of square roots, and simplify further to arrive at this result.
Q: How do I simplify $\sqrt{a^5}$?
A: To simplify $\sqrt{a^5}$, we can break down the expression into its prime factors. We can express $a^5$ as $a^4 \cdot a$. Then, we can apply the properties of square roots to simplify further. The result is $a^2 \sqrt{a}$.
Q: What is the simplest form of $\sqrt{a^6}$?
A: The simplest form of $\sqrt{a^6}$ is $a^3$. This is because we can break down the expression into its prime factors, apply the properties of square roots, and simplify further to arrive at this result.
Q: How do I simplify $\sqrt{a^9}$?
A: To simplify $\sqrt{a^9}$, we can break down the expression into its prime factors. We can express $a^9$ as $a^8 \cdot a$. Then, we can apply the properties of square roots to simplify further. The result is $a^4 \sqrt{a}$.
Q: What is the simplest form of $\sqrt{a^0}$?
A: The simplest form of $\sqrt{a^0}$ is 1. This is because any number raised to the power of 0 is equal to 1.
Q: How do I simplify $\sqrt{a^1}$?
A: To simplify $\sqrt{a^1}$, we can break down the expression into its prime factors. We can express $a^1$ as $a$. Then, we can apply the properties of square roots to simplify further. The result is $\sqrt{a}$.
Conclusion
In conclusion, simplifying square roots involves breaking down the expression into its prime factors, applying the properties of square roots, and simplifying further. By following these steps, we can arrive at the simplest form of any square root expression.
Tips and Tricks
- Always break down the expression into its prime factors before simplifying.
- Apply the properties of square roots to simplify further.
- Simplify the expression as much as possible to arrive at the simplest form.
Final Answer
The final answer is that simplifying square roots involves breaking down the expression into its prime factors, applying the properties of square roots, and simplifying further. By following these steps, we can arrive at the simplest form of any square root expression.