Simplify The Expression: $\[ 3a^{\frac{1}{2}}b^2 + 4a^{-3a}b^2 \\]

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us to solve complex problems and understand the underlying concepts. In this article, we will focus on simplifying the given expression: ${ 3a{\frac{1}{2}}b2 + 4a{-3a}b2 }$. We will break down the expression into smaller parts, apply the rules of exponents, and simplify it step by step.

Understanding the Expression

The given expression is a combination of two terms: 3a12b23a^{\frac{1}{2}}b^2 and 4aβˆ’3ab24a^{-3a}b^2. To simplify this expression, we need to understand the rules of exponents and how to handle negative exponents.

Rules of Exponents

Before we dive into simplifying the expression, let's review the rules of exponents:

  • Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, amβ‹…an=am+na^m \cdot a^n = a^{m+n}.
  • Power of a Power: When raising a power to another power, we multiply the exponents. For example, (am)n=amn(a^m)^n = a^{mn}.
  • Negative Exponent: A negative exponent indicates that we need to take the reciprocal of the base. For example, aβˆ’m=1ama^{-m} = \frac{1}{a^m}.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's simplify the given expression.

Step 1: Simplify the First Term

The first term is 3a12b23a^{\frac{1}{2}}b^2. We can simplify this term by applying the rule of exponents for a product of powers:

3a12b2=3β‹…a12β‹…b23a^{\frac{1}{2}}b^2 = 3 \cdot a^{\frac{1}{2}} \cdot b^2

Since a12a^{\frac{1}{2}} is the same as a\sqrt{a}, we can rewrite the term as:

3a12b2=3ab23a^{\frac{1}{2}}b^2 = 3\sqrt{a}b^2

Step 2: Simplify the Second Term

The second term is 4aβˆ’3ab24a^{-3a}b^2. We can simplify this term by applying the rule of exponents for a negative exponent:

4aβˆ’3ab2=4β‹…1a3aβ‹…b24a^{-3a}b^2 = 4 \cdot \frac{1}{a^{3a}} \cdot b^2

Since 1a3a\frac{1}{a^{3a}} is the same as aβˆ’3aa^{-3a}, we can rewrite the term as:

4aβˆ’3ab2=4b2a3a4a^{-3a}b^2 = \frac{4b^2}{a^{3a}}

Step 3: Combine the Terms

Now that we have simplified both terms, we can combine them:

3a12b2+4aβˆ’3ab2=3ab2+4b2a3a3a^{\frac{1}{2}}b^2 + 4a^{-3a}b^2 = 3\sqrt{a}b^2 + \frac{4b^2}{a^{3a}}

Step 4: Simplify the Expression Further

We can simplify the expression further by combining the two terms:

3ab2+4b2a3a=3ab2a3a+4b2a3a3\sqrt{a}b^2 + \frac{4b^2}{a^{3a}} = \frac{3\sqrt{a}b^2a^{3a} + 4b^2}{a^{3a}}

Since a3aa^{3a} is the same as a3aa^{3a}, we can rewrite the expression as:

3ab2a3a+4b2a3a=3ab2a3a+4b2a3a\frac{3\sqrt{a}b^2a^{3a} + 4b^2}{a^{3a}} = \frac{3\sqrt{a}b^2a^{3a} + 4b^2}{a^{3a}}

Step 5: Final Simplification

We can simplify the expression further by factoring out the common term b2b^2:

3ab2a3a+4b2a3a=b2(3aa3a+4a3a)\frac{3\sqrt{a}b^2a^{3a} + 4b^2}{a^{3a}} = b^2 \left( \frac{3\sqrt{a}a^{3a} + 4}{a^{3a}} \right)

Since 3aa3a+4a3a\frac{3\sqrt{a}a^{3a} + 4}{a^{3a}} is the same as 3aa3aa3a+4a3a\frac{3\sqrt{a}a^{3a}}{a^{3a}} + \frac{4}{a^{3a}}, we can rewrite the expression as:

b2(3aa3a+4a3a)=b2(3a+4a3a)b^2 \left( \frac{3\sqrt{a}a^{3a} + 4}{a^{3a}} \right) = b^2 \left( 3\sqrt{a} + \frac{4}{a^{3a}} \right)

Conclusion

In this article, we simplified the given expression ${ 3a{\frac{1}{2}}b2 + 4a{-3a}b2 }$ step by step. We applied the rules of exponents, simplified the terms, and combined them to get the final simplified expression. We hope that this article has helped you to understand the process of simplifying expressions and how to apply the rules of exponents.

Final Answer

The final simplified expression is:

b^2 \left( 3\sqrt{a} + \frac{4}{a^{3a}} \right)$<br/> **Simplify the Expression: A Comprehensive Guide** ===================================================== **Q&A: Simplifying Expressions** ----------------------------- In the previous article, we simplified the given expression ${ 3a^{\frac{1}{2}}b^2 + 4a^{-3a}b^2 }$ step by step. However, we understand that simplifying expressions can be a challenging task, and you may have some questions. In this article, we will answer some of the most frequently asked questions about simplifying expressions. **Q: What are the rules of exponents?** -------------------------------------- A: The rules of exponents are a set of rules that help us to simplify expressions with exponents. The three main rules of exponents are: * **Product of Powers**: When multiplying two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$. * **Power of a Power**: When raising a power to another power, we multiply the exponents. For example, $(a^m)^n = a^{mn}$. * **Negative Exponent**: A negative exponent indicates that we need to take the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$. **Q: How do I simplify an expression with a negative exponent?** --------------------------------------------------------- A: To simplify an expression with a negative exponent, we need to take the reciprocal of the base. For example, if we have the expression $a^{-m}$, we can simplify it by taking the reciprocal of the base: $a^{-m} = \frac{1}{a^m}

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, we need to use the rule of exponents for a product of powers. For example, if we have the expression amna^{\frac{m}{n}}, we can simplify it by adding the exponents:

amn=(am)1n=amna^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = a^{\frac{m}{n}}

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent. However, you need to be careful when simplifying expressions with variables in the exponent. For example, if we have the expression a2xa^{2x}, we can simplify it by using the rule of exponents for a power of a power:

(ax)2=a2x(a^x)^2 = a^{2x}

Q: How do I simplify an expression with multiple terms?

A: To simplify an expression with multiple terms, you need to combine the terms using the rules of exponents. For example, if we have the expression am+ana^m + a^n, we can simplify it by combining the terms:

am+an=am+na^m + a^n = a^{m+n}

Q: Can I simplify an expression with a coefficient?

A: Yes, you can simplify an expression with a coefficient. However, you need to be careful when simplifying expressions with coefficients. For example, if we have the expression 2am2a^m, we can simplify it by using the rule of exponents for a product of powers:

2am=2β‹…am2a^m = 2 \cdot a^m

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying expressions. We hope that this article has helped you to understand the process of simplifying expressions and how to apply the rules of exponents.

Final Answer

The final simplified expression is:

b2(3a+4a3a)b^2 \left( 3\sqrt{a} + \frac{4}{a^{3a}} \right)