Simplify The Following Expressions:1. $2 \sqrt{32 X^2 Y^3} - X Y \sqrt{98 Y}$2. 9 X ⋅ 4 X 7 \sqrt{9 X} \cdot \sqrt{4 X^7} 9 X ​ ⋅ 4 X 7 ​

Introduction

Mathematical expressions can be complex and difficult to understand, especially when they involve square roots and variables. In this article, we will simplify two given mathematical expressions using algebraic manipulation and properties of square roots.

Expression 1: Simplify $2 \sqrt{32 x^2 y^3} - x y \sqrt{98 y}$

Step 1: Factorize the Numbers Inside the Square Roots

The first step in simplifying the expression is to factorize the numbers inside the square roots. We can factorize 32 as $2^5$ and 98 as $2 \cdot 7^2$.

2 \sqrt{32 x^2 y^3} - x y \sqrt{98 y} = 2 \sqrt{2^5 x^2 y^3} - x y \sqrt{2 \cdot 7^2 y}

Step 2: Simplify the Square Roots

Now, we can simplify the square roots by taking out the perfect squares.

2 \sqrt{2^5 x^2 y^3} - x y \sqrt{2 \cdot 7^2 y} = 2 \cdot 2^2 \sqrt{x^2 y^3} - x y \sqrt{2} \cdot 7 \sqrt{y}

Step 3: Simplify the Variables Inside the Square Roots

Next, we can simplify the variables inside the square roots by taking out the perfect squares.

2 \cdot 2^2 \sqrt{x^2 y^3} - x y \sqrt{2} \cdot 7 \sqrt{y} = 4 x \sqrt{y^3} - 7 x y \sqrt{y}

Step 4: Simplify the Variables Inside the Square Roots (continued)

We can further simplify the variables inside the square roots by taking out the perfect squares.

4 x \sqrt{y^3} - 7 x y \sqrt{y} = 4 x y \sqrt{y^2} - 7 x y \sqrt{y}

Step 5: Simplify the Square Roots (continued)

Now, we can simplify the square roots by taking out the perfect squares.

4 x y \sqrt{y^2} - 7 x y \sqrt{y} = 4 x y y - 7 x y \sqrt{y}

Step 6: Simplify the Expression

Finally, we can simplify the expression by combining like terms.

4 x y y - 7 x y \sqrt{y} = 4 x y^2 - 7 x y \sqrt{y}

Expression 2: Simplify $\sqrt{9 x} \cdot \sqrt{4 x^7}$

Step 1: Simplify the Square Roots

The first step in simplifying the expression is to simplify the square roots.

\sqrt{9 x} \cdot \sqrt{4 x^7} = \sqrt{3^2 x} \cdot \sqrt{2^2 x^7}

Step 2: Simplify the Square Roots (continued)

Now, we can simplify the square roots by taking out the perfect squares.

\sqrt{3^2 x} \cdot \sqrt{2^2 x^7} = 3 \sqrt{x} \cdot 2 \sqrt{x^7}

Step 3: Simplify the Variables Inside the Square Roots

Next, we can simplify the variables inside the square roots by taking out the perfect squares.

3 \sqrt{x} \cdot 2 \sqrt{x^7} = 3 \cdot 2 \sqrt{x^8}

Step 4: Simplify the Expression

Finally, we can simplify the expression by combining like terms.

3 \cdot 2 \sqrt{x^8} = 6 x^4

Conclusion

Introduction

In our previous article, we simplified two given mathematical expressions using algebraic manipulation and properties of square roots. In this article, we will answer some frequently asked questions related to the simplification of mathematical expressions.

Q: What is the first step in simplifying a mathematical expression?

A: The first step in simplifying a mathematical expression is to factorize the numbers inside the square roots. This involves breaking down the numbers into their prime factors.

Q: How do I simplify a square root with a variable inside it?

A: To simplify a square root with a variable inside it, you need to take out the perfect squares. This involves identifying the perfect squares inside the square root and taking them out.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, 4 is a perfect square because it can be expressed as 2^2. A non-perfect square is a number that cannot be expressed as the square of an integer.

Q: How do I simplify a product of two square roots?

A: To simplify a product of two square roots, you need to multiply the numbers inside the square roots and then simplify the result.

Q: What is the property of square roots that allows us to simplify expressions?

A: The property of square roots that allows us to simplify expressions is the property that states that the square root of a product is equal to the product of the square roots. This property is expressed mathematically as:

√(ab) = √a√b

Q: Can I simplify an expression with a negative number inside the square root?

A: Yes, you can simplify an expression with a negative number inside the square root. However, you need to remember that the square root of a negative number is an imaginary number.

Q: How do I simplify an expression with a variable inside the square root that has a negative exponent?

A: To simplify an expression with a variable inside the square root that has a negative exponent, you need to take out the perfect squares and then simplify the result.

Q: What is the final step in simplifying a mathematical expression?

A: The final step in simplifying a mathematical expression is to combine like terms. This involves combining the terms that have the same variable and exponent.

Example Questions

Question 1

Simplify the expression: √(16x^2y3) - xy√(98y)

Solution

To simplify the expression, we need to factorize the numbers inside the square roots and then simplify the result.

√(16x^2y3) - xy√(98y) = √(2^4x2y^3) - xy√(2 \cdot 7^2y)

= 2^2√(x2y^3) - xy√(2) \cdot 7√y

= 4x√(y^3) - 7xy√y

= 4xy√(y^2) - 7xy√y

= 4xy^2 - 7xy√y

Question 2

Simplify the expression: √(9x) \cdot √(4x^7)

Solution

To simplify the expression, we need to multiply the numbers inside the square roots and then simplify the result.

√(9x) \cdot √(4x^7) = √(3^2x) \cdot √(2^2x7)

= 3√x \cdot 2√(x^7)

= 3 \cdot 2√(x^8)

= 6x^4

Conclusion

In this article, we answered some frequently asked questions related to the simplification of mathematical expressions. We also provided example questions and solutions to help illustrate the concepts.