Simplify: √(8x⁷u²) √(2x²u) Assume That All Variables Represent Positive Real Numbers.

May 4, 2025 by ADMIN 86 views

Simplify: √(8x⁷u²) √(2x²u)

In this article, we will simplify the given expression √(8x⁷u²) √(2x²u) under the assumption that all variables represent positive real numbers. We will use the properties of radicals and exponents to simplify the expression.

Understanding the Expression

The given expression is the product of two square roots:

√(8x⁷u²) √(2x²u)

To simplify this expression, we need to understand the properties of radicals and exponents. We know that the square root of a number is a value that, when multiplied by itself, gives the original number. For example, √4 = 2, because 2 × 2 = 4.

Properties of Radicals

There are several properties of radicals that we will use to simplify the expression:

Product of Radicals: The product of two or more radicals is equal to the product of the numbers inside the radicals.
Power of a Radical: When a radical is raised to a power, the power can be brought down as an exponent.
Multiplication of Radicals: When two or more radicals are multiplied together, the product can be simplified by multiplying the numbers inside the radicals.

Simplifying the Expression

Now that we have understood the properties of radicals, let's simplify the given expression:

√(8x⁷u²) √(2x²u)

Using the property of product of radicals, we can write:

√(8x⁷u²) √(2x²u) = √(8 × 2 × x⁷ × x² × u² × u)

Simplifying the numbers inside the radical, we get:

√(16x⁹u³)

Now, we can use the property of power of a radical to bring down the exponent:

√(16x⁹u³) = √(16) × √(x⁹) × √(u³)

Simplifying further, we get:

4x³√(x)u√(u)

In this article, we simplified the given expression √(8x⁷u²) √(2x²u) using the properties of radicals and exponents. We used the product of radicals, power of a radical, and multiplication of radicals properties to simplify the expression. The final simplified expression is 4x³√(x)u√(u).

Final Answer

The final answer is 4x³√(x)u√(u).

Step-by-Step Solution

Here are the step-by-step solutions to simplify the expression:

Use the property of product of radicals to write √(8x⁷u²) √(2x²u) = √(8 × 2 × x⁷ × x² × u² × u)
Simplify the numbers inside the radical: √(8 × 2 × x⁷ × x² × u² × u) = √(16x⁹u³)
Use the property of power of a radical to bring down the exponent: √(16x⁹u³) = √(16) × √(x⁹) × √(u³)
Simplify further: √(16) × √(x⁹) × √(u³) = 4x³√(x)u√(u)

Common Mistakes

Here are some common mistakes to avoid when simplifying radicals:

Not using the property of product of radicals: When simplifying the product of two or more radicals, make sure to use the property of product of radicals.
Not using the property of power of a radical: When a radical is raised to a power, make sure to use the property of power of a radical to bring down the exponent.
Not simplifying the numbers inside the radical: Make sure to simplify the numbers inside the radical before simplifying the expression.

Real-World Applications

Simplifying radicals has many real-world applications in mathematics and science. Here are a few examples:

Physics: In physics, radicals are used to represent the square root of a quantity. For example, the speed of an object can be represented as √(v² + w²), where v and w are the components of the velocity.
Engineering: In engineering, radicals are used to represent the square root of a quantity. For example, the stress on a beam can be represented as √(σ² + τ²), where σ and τ are the components of the stress.
Computer Science: In computer science, radicals are used to represent the square root of a quantity. For example, the distance between two points in a graph can be represented as √(x² + y²), where x and y are the coordinates of the points.

In our previous article, we simplified the given expression √(8x⁷u²) √(2x²u) using the properties of radicals and exponents. In this article, we will answer some frequently asked questions related to simplifying radicals.

Q: What is the property of product of radicals?

A: The property of product of radicals states that the product of two or more radicals is equal to the product of the numbers inside the radicals. For example, √(a) √(b) = √(ab).

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to use the properties of radicals, such as the product of radicals, power of a radical, and multiplication of radicals. You also need to simplify the numbers inside the radical before simplifying the expression.

Q: What is the difference between a radical and an exponent?

A: A radical and an exponent are both used to represent a quantity that is raised to a power. However, a radical is used to represent the square root of a quantity, while an exponent is used to represent a quantity that is raised to a power.

Q: Can I simplify a radical expression with a negative exponent?

A: Yes, you can simplify a radical expression with a negative exponent. To do this, you need to use the property of power of a radical to bring down the exponent. For example, √(x⁻²) = 1/x√(x).

Q: How do I simplify a radical expression with a variable in the radicand?

A: To simplify a radical expression with a variable in the radicand, you need to use the property of product of radicals to simplify the expression. For example, √(x²y) = xy√(x).

Q: Can I simplify a radical expression with a fraction in the radicand?

A: Yes, you can simplify a radical expression with a fraction in the radicand. To do this, you need to use the property of product of radicals to simplify the expression. For example, √(x/2) = √(x)/√(2).

Q: How do I simplify a radical expression with a complex number in the radicand?

A: To simplify a radical expression with a complex number in the radicand, you need to use the property of product of radicals to simplify the expression. For example, √(3 + 4i) = √(3) + √(4i).

In conclusion, simplifying radicals is an important concept in mathematics and science. By understanding the properties of radicals and using them to simplify expressions, we can solve complex problems and represent quantities in a more concise and elegant way. We hope that this Q&A article has helped you to understand the concept of simplifying radicals better.

Common Mistakes

Here are some common mistakes to avoid when simplifying radicals:

Not using the property of product of radicals: When simplifying the product of or more radicals, make sure to use the property of product of radicals.
Not using the property of power of a radical: When a radical is raised to a power, make sure to use the property of power of a radical to bring down the exponent.
Not simplifying the numbers inside the radical: Make sure to simplify the numbers inside the radical before simplifying the expression.

Real-World Applications

Simplifying radicals has many real-world applications in mathematics and science. Here are a few examples:

Physics: In physics, radicals are used to represent the square root of a quantity. For example, the speed of an object can be represented as √(v² + w²), where v and w are the components of the velocity.
Engineering: In engineering, radicals are used to represent the square root of a quantity. For example, the stress on a beam can be represented as √(σ² + τ²), where σ and τ are the components of the stress.
Computer Science: In computer science, radicals are used to represent the square root of a quantity. For example, the distance between two points in a graph can be represented as √(x² + y²), where x and y are the coordinates of the points.