Simplify The Expression: ${ A \sqrt{5x^2 + 4x} }$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it often involves manipulating square roots, fractions, and other mathematical operations. In this article, we will focus on simplifying the given expression: ${ a \sqrt{5x^2 + 4x} }$. This expression involves a square root, and our goal is to simplify it by factoring and manipulating the terms inside the square root.

Understanding the Expression

The given expression is ${ a \sqrt{5x^2 + 4x} }$. To simplify this expression, we need to understand the properties of square roots and how to manipulate them. The 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.

Factoring the Expression

To simplify the given expression, we need to factor the terms inside the square root. Factoring involves expressing a polynomial as a product of simpler polynomials. In this case, we can factor the expression $5x^2 + 4x$ by taking out the greatest common factor (GCF). The GCF of $5x^2$ and $4x$ is $x$, so we can factor the expression as follows:

$${ 5x^2 + 4x = x(5x + 4) }$$

Simplifying the Expression

Now that we have factored the expression inside the square root, we can simplify the original expression. We can rewrite the expression as follows:

$${ a \sqrt{5x^2 + 4x} = a \sqrt{x(5x + 4)} }$$

Using the Property of Square Roots

The property of square roots states that the square root of a product is equal to the product of the square roots. In other words, ${ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} }$. We can use this property to simplify the expression further:

$${ a \sqrt{x(5x + 4)} = a \sqrt{x} \cdot \sqrt{5x + 4} }$$

Simplifying the Expression Further

We can simplify the expression further by combining the terms inside the square roots. We can rewrite the expression as follows:

$${ a \sqrt{x} \cdot \sqrt{5x + 4} = a \sqrt{5x^2 + 4x} }$$

Conclusion

In this article, we simplified the given expression ${ a \sqrt{5x^2 + 4x} }$. We factored the terms inside the square root and used the property of square roots to simplify the expression. The final simplified expression is ${ a \sqrt{5x^2 + 4x} }$. This expression involves a square root, and we were able to simplify it by factoring and manipulating the terms inside the square root.

Frequently Asked Questions

• Q: What is the property of square roots? A: The property of square roots states that the square root of a product is equal to the product of the square roots. In other words, ${ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} }$.
• Q: How do I simplify an expression involving a square root? A: To simplify an expression involving a square root, you need to factor the terms inside the square root and use the property of square roots to manipulate the expression.
• Q: What is the final simplified expression? A: The final simplified expression is ${ a \sqrt{5x^2 + 4x} }$.

Final Answer

The final answer is ${ a \sqrt{5x^2 + 4x} }$.

Introduction

In our previous article, we simplified the given expression ${ a \sqrt{5x^2 + 4x} }$. We factored the terms inside the square root and used the property of square roots to manipulate the expression. In this article, we will answer some frequently asked questions related to simplifying expressions involving square roots.

Q&A

Q: What is the property of square roots?

A: The property of square roots states that the square root of a product is equal to the product of the square roots. In other words, ${ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} }$. This property is useful when simplifying expressions involving square roots.

Q: How do I simplify an expression involving a square root?

A: To simplify an expression involving a square root, you need to factor the terms inside the square root and use the property of square roots to manipulate the expression. You can also use other mathematical operations such as addition, subtraction, multiplication, and division to simplify the expression.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides two or more numbers without leaving a remainder. In the context of simplifying expressions involving square roots, the GCF is used to factor the terms inside the square root.

Q: How do I find the GCF of two or more numbers?

A: To find the GCF of two or more numbers, you need to list the factors of each number and find the largest factor that is common to all the numbers. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The greatest common factor of 12 and 18 is 6.

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

A: A square root and a radical are both mathematical operations that involve finding the value of a number that, when multiplied by itself, gives the original number. However, a square root is a specific type of radical that involves finding the value of a number that, when multiplied by itself, gives a perfect square. A radical, on the other hand, is a more general term that includes square roots as well as other types of roots.

Q: How do I simplify an expression involving a square root and a fraction?

A: To simplify an expression involving a square root and a fraction, you need to use the property of square roots and the rules of fractions to manipulate the expression. You can also use other mathematical operations such as addition, subtraction, multiplication, and division to simplify the expression.

Q: What is the final simplified expression?

A: The final simplified expression is ${ a \sqrt{5x^2 + 4x} }$. This expression involves a square root, and we were able to simplify it by factoring and manipulating the terms inside the square root.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions involving square roots. We discussed the property of square roots, the greatest common factor, and how to simplify expressions involving square roots and fractions. We also provided some examples and explanations to help illustrate the concepts.

Frequently Asked Questions

• Q: What is the property of square roots? A: The property of square roots states that the square root of a product is equal to the product of the square roots. In other words, ${ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} }$.
• Q: How do I simplify an expression involving a square root? A: To simplify an expression involving a square root, you need to factor the terms inside the square root and use the property of square roots to manipulate the expression.
• Q: What is the greatest common factor (GCF)? A: The greatest common factor (GCF) is the largest factor that divides two or more numbers without leaving a remainder.
• Q: How do I find the GCF of two or more numbers? A: To find the GCF of two or more numbers, you need to list the factors of each number and find the largest factor that is common to all the numbers.

Final Answer

The final answer is ${ a \sqrt{5x^2 + 4x} }$.