Simplify The Expression:${ A \sqrt{32 A^2 B^3} + \sqrt{98 A^5 B^2} + B \sqrt{50 A^4 B} }$

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Introduction

In this article, we will delve into the world of algebra and simplify a given expression involving square roots. The expression is a combination of terms with square roots, and our goal is to simplify it to its most basic form. We will use various mathematical techniques and formulas to achieve this goal.

Understanding the Expression

The given expression is:

a32a2b3+98a5b2+b50a4b{ a \sqrt{32 a^2 b^3} + \sqrt{98 a^5 b^2} + b \sqrt{50 a^4 b} }

This expression consists of three terms, each involving a square root. To simplify this expression, we need to first understand the properties of square roots and how to manipulate them.

Properties of Square Roots

Before we proceed with simplifying the expression, let's review some important properties of square roots:

  • Square Root of a Product: ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
  • Square Root of a Power: an=an/2\sqrt{a^n} = a^{n/2}
  • Square Root of a Constant: c=c1/2\sqrt{c} = c^{1/2}

These properties will be essential in simplifying the given expression.

Simplifying the First Term

The first term in the expression is:

a32a2b3{ a \sqrt{32 a^2 b^3} }

Using the property of square roots of a product, we can rewrite this term as:

a32a2b3=a32⋅a2⋅b3{ a \sqrt{32 a^2 b^3} = a \sqrt{32} \cdot \sqrt{a^2} \cdot \sqrt{b^3} }

Now, we can simplify the square roots:

a32⋅a2⋅b3=a⋅42⋅a⋅bb{ a \sqrt{32} \cdot \sqrt{a^2} \cdot \sqrt{b^3} = a \cdot 4 \sqrt{2} \cdot a \cdot b \sqrt{b} }

Combining like terms, we get:

a⋅42⋅a⋅bb=4a2b2b{ a \cdot 4 \sqrt{2} \cdot a \cdot b \sqrt{b} = 4 a^2 b \sqrt{2 b} }

Simplifying the Second Term

The second term in the expression is:

98a5b2{ \sqrt{98 a^5 b^2} }

Using the property of square roots of a product, we can rewrite this term as:

98a5b2=98⋅a5⋅b2{ \sqrt{98 a^5 b^2} = \sqrt{98} \cdot \sqrt{a^5} \cdot \sqrt{b^2} }

Now, we can simplify the square roots:

98⋅a5⋅b2=72⋅a5/2⋅b{ \sqrt{98} \cdot \sqrt{a^5} \cdot \sqrt{b^2} = 7 \sqrt{2} \cdot a^{5/2} \cdot b }

Combining like terms, we get:

72⋅a5/2⋅b=7a5/2b2{ 7 \sqrt{2} \cdot a^{5/2} \cdot b = 7 a^{5/2} b \sqrt{2} }

Simplifying the Third Term

The third term in the expression is:

b50a4b{ b \sqrt{50 a^4 b} }

Using the property of square roots of a product, we can rewrite this term as:

b50a4b=b50⋅a4⋅b{ b \sqrt{50 a^4 b} = b \sqrt{50} \cdot \sqrt{a^4} \cdot \sqrt{b} }

Now, we can simplify the square roots:

b50⋅a4⋅b=b⋅52⋅a2⋅b{ b \sqrt{50} \cdot \sqrt{a^4} \cdot \sqrt{b} = b \cdot 5 \sqrt{2} \cdot a^2 \cdot \sqrt{b} }

Combining like terms, we get:

b⋅52⋅a2⋅b=5a2b2b{ b \cdot 5 \sqrt{2} \cdot a^2 \cdot \sqrt{b} = 5 a^2 b \sqrt{2 b} }

Combining the Terms

Now that we have simplified each term, we can combine them to get the final simplified expression:

4a2b2b+7a5/2b2+5a2b2b{ 4 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} + 5 a^2 b \sqrt{2 b} }

Combining like terms, we get:

4a2b2b+5a2b2b+7a5/2b2{ 4 a^2 b \sqrt{2 b} + 5 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} }

=9a2b2b+7a5/2b2{ = 9 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} }

Conclusion

In this article, we simplified a given expression involving square roots. We used various mathematical techniques and formulas to achieve this goal. The final simplified expression is:

9a2b2b+7a5/2b2{ 9 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} }

This expression is now in its most basic form, and we can use it for further calculations or applications.

Final Answer

The final answer is:

Q&A: Simplifying the Expression

In this article, we will answer some frequently asked questions related to simplifying the expression:

a32a2b3+98a5b2+b50a4b{ a \sqrt{32 a^2 b^3} + \sqrt{98 a^5 b^2} + b \sqrt{50 a^4 b} }

Q: What are the properties of square roots that are used to simplify the expression?

A: The properties of square roots used to simplify the expression are:

  • Square Root of a Product: ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
  • Square Root of a Power: an=an/2\sqrt{a^n} = a^{n/2}
  • Square Root of a Constant: c=c1/2\sqrt{c} = c^{1/2}

Q: How do you simplify the first term in the expression?

A: To simplify the first term, we use the property of square roots of a product to rewrite it as:

a32a2b3=a32⋅a2⋅b3{ a \sqrt{32 a^2 b^3} = a \sqrt{32} \cdot \sqrt{a^2} \cdot \sqrt{b^3} }

Then, we simplify the square roots:

a32⋅a2⋅b3=a⋅42⋅a⋅bb{ a \sqrt{32} \cdot \sqrt{a^2} \cdot \sqrt{b^3} = a \cdot 4 \sqrt{2} \cdot a \cdot b \sqrt{b} }

Combining like terms, we get:

a⋅42⋅a⋅bb=4a2b2b{ a \cdot 4 \sqrt{2} \cdot a \cdot b \sqrt{b} = 4 a^2 b \sqrt{2 b} }

Q: How do you simplify the second term in the expression?

A: To simplify the second term, we use the property of square roots of a product to rewrite it as:

98a5b2=98⋅a5⋅b2{ \sqrt{98 a^5 b^2} = \sqrt{98} \cdot \sqrt{a^5} \cdot \sqrt{b^2} }

Then, we simplify the square roots:

98⋅a5⋅b2=72⋅a5/2⋅b{ \sqrt{98} \cdot \sqrt{a^5} \cdot \sqrt{b^2} = 7 \sqrt{2} \cdot a^{5/2} \cdot b }

Combining like terms, we get:

72⋅a5/2⋅b=7a5/2b2{ 7 \sqrt{2} \cdot a^{5/2} \cdot b = 7 a^{5/2} b \sqrt{2} }

Q: How do you simplify the third term in the expression?

A: To simplify the third term, we use the property of square roots of a product to rewrite it as:

b50a4b=b50⋅a4⋅b{ b \sqrt{50 a^4 b} = b \sqrt{50} \cdot \sqrt{a^4} \cdot \sqrt{b} }

Then, we simplify the square roots:

b50⋅a4⋅b=b⋅52⋅a2⋅b{ b \sqrt{50} \cdot \sqrt{a^4} \cdot \sqrt{b} = b \cdot 5 \sqrt{2} \cdot a^2 \cdot \sqrt{b} }

Combining like terms, we get:

b⋅52⋅a2⋅b=5a2b2b{ b \cdot 5 \sqrt{2} \cdot a^2 \cdot \sqrt{b} = 5 a^2 b \sqrt{2 b} }

Q: How do you combine the simplified terms to get the final expression?

A: To combine the simplified terms, we add them together:

4a2b2b+7a5/2b2+5a2b2b{ 4 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} + 5 a^2 b \sqrt{2 b} }

Combining like terms, we get:

4a2b2b+5a2b2b+7a5/2b2{ 4 a^2 b \sqrt{2 b} + 5 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} }

=9a2b2b+7a5/2b2{ = 9 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} }

Q: What is the final simplified expression?

A: The final simplified expression is:

9a2b2b+7a5/2b2{ 9 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} }

Conclusion

In this article, we answered some frequently asked questions related to simplifying the expression:

a32a2b3+98a5b2+b50a4b{ a \sqrt{32 a^2 b^3} + \sqrt{98 a^5 b^2} + b \sqrt{50 a^4 b} }

We used various mathematical techniques and formulas to simplify the expression and arrived at the final simplified expression:

9a2b2b+7a5/2b2{ 9 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} }

This expression is now in its most basic form, and we can use it for further calculations or applications.

Final Answer

The final answer is:

9a2b2b+7a5/2b2{ 9 a^2 b \sqrt{2 b} + 7 a^{5/2} b \sqrt{2} }