If $x\ \textgreater \ 0$, What Is The Product Of $7 \sqrt{5 X^3} \cdot 9 X \sqrt{24 X}$ In Simplest Radical Form?

Understanding the Problem

When dealing with expressions involving square roots, it's essential to simplify them to their simplest radical form. This involves expressing the square root of a product as the product of square roots, and then simplifying the resulting expression.

Simplifying the Expression

To simplify the given expression, we need to start by expressing the square roots as products of square roots.

$$7 \sqrt{5 x^3} \cdot 9 x \sqrt{24 x}$$

We can rewrite the expression as:

$$7 \cdot 9 \cdot x \cdot \sqrt{5 x^3} \cdot \sqrt{24 x}$$

Simplifying the Square Roots

Now, we can simplify the square roots by expressing them as products of square roots.

$$\sqrt{5 x^3} \cdot \sqrt{24 x}$$

We can rewrite the expression as:

$$\sqrt{5 x^3 \cdot 24 x}$$

Simplifying the Product Under the Square Root

Now, we can simplify the product under the square root by multiplying the numbers and combining the variables.

$$5 x^3 \cdot 24 x$$

We can rewrite the expression as:

$$120 x^4$$

Simplifying the Square Root

Now, we can simplify the square root by expressing it as a product of square roots.

$$\sqrt{120 x^4}$$

We can rewrite the expression as:

$$\sqrt{120} \cdot \sqrt{x^4}$$

Simplifying the Square Root of 120

Now, we can simplify the square root of 120 by expressing it as a product of prime factors.

$$\sqrt{120}$$

We can rewrite the expression as:

$$\sqrt{2^2 \cdot 3 \cdot 5}$$

We can simplify the expression as:

$$2 \sqrt{15}$$

Simplifying the Square Root of x^4

Now, we can simplify the square root of x^4 by expressing it as a product of square roots.

$$\sqrt{x^4}$$

We can rewrite the expression as:

$$x^2$$

Combining the Simplified Expressions

Now, we can combine the simplified expressions to get the final result.

$$7 \cdot 9 \cdot x \cdot 2 \sqrt{15} \cdot x^2$$

We can rewrite the expression as:

$$126 x^3 \sqrt{15}$$

Conclusion

In this article, we have simplified the given expression to its simplest radical form. We started by expressing the square roots as products of square roots, and then simplified the resulting expression by combining the variables and multiplying the numbers. The final result is:

$$126 x^3 \sqrt{15}$$

This is the simplest radical form of the given expression.

Understanding the Basics

In our previous article, we simplified the expression $7 \sqrt{5 x^3} \cdot 9 x \sqrt{24 x}$ to its simplest radical form. However, we received many questions from readers who were unsure about the steps involved in simplifying expressions with square roots. In this article, we will answer some of the most frequently asked questions about simplifying expressions with square roots.

Q: What is the difference between a square root and a radical?

A: A square root and a radical are often used interchangeably, but technically, a square root is a specific type of radical. A radical is a mathematical expression that represents the product of a number and itself, such as $\sqrt{16}$ or $\sqrt{25}$. A square root is a type of radical that represents the product of a number and itself, where the number is a perfect square.

Q: How do I simplify a square root expression?

A: To simplify a square root expression, you need to follow these steps:

1. Express the square root as a product of square roots.
2. Simplify the product under the square root by multiplying the numbers and combining the variables.
3. Simplify the square root of each factor by expressing it as a product of prime factors.
4. Combine the simplified expressions to get the final result.

Q: What is the rule for simplifying square roots of products?

A: The rule for simplifying square roots of products is:

$$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$

This means that you can simplify the square root of a product by expressing it as the product of square roots.

Q: How do I simplify a square root of a fraction?

A: To simplify a square root of a fraction, you need to follow these steps:

1. Express the fraction as a product of prime factors.
2. Simplify the square root of each factor by expressing it as a product of prime factors.
3. Combine the simplified expressions to get the final result.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a ratio of two integers, such as $\frac{3}{4}$ or $\frac{5}{6}$. An irrational number is a number that cannot be expressed as a ratio of two integers, such as $\sqrt{2}$ or $\pi$.

Q: How do I determine if a number is rational or irrational?

A: To determine if a number is rational or irrational, you need to follow these steps:

1. Express the number as a decimal or a fraction.
2. Check if the decimal or fraction can be expressed as a ratio of two integers.
3. If it can be expressed as a ratio of two integers, then it is a rational number. If it cannot be expressed as a ratio of two integers, then it is an irrational number.

Q: What is the significance of simplifying expressions with square roots?

A: Simplifying expressions with square roots is important because it helps to:

1. Reduce the complexity of the expression.
2. Make it easier to solve equations and inequalities.
3. Improve the accuracy of calculations.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying expressions with square roots. We have covered topics such as the difference between a square root and a radical, the rule for simplifying square roots of products, and the significance of simplifying expressions with square roots. We hope that this article has been helpful in clarifying any doubts you may have had about simplifying expressions with square roots.