Simplify The Expression: $\left(15 A^3 B\right) \div \left(3 A B^2\right$\]

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with algebraic expressions, we often need to simplify them to make them easier to work with. In this article, we will focus on simplifying the given expression (15a3b)÷(3ab2)\left(15 a^3 b\right) \div \left(3 a b^2\right) using the rules of exponents and division.

Understanding the Expression

The given expression is a division of two algebraic expressions: (15a3b)\left(15 a^3 b\right) and (3ab2)\left(3 a b^2\right). To simplify this expression, we need to apply the rules of exponents and division. The expression can be rewritten as 15a3b3ab2\frac{15 a^3 b}{3 a b^2}.

Applying the Rules of Exponents

When dividing two algebraic expressions with the same base, we subtract the exponents. In this case, the base is aa and bb. We can rewrite the expression as 153⋅a3−1a1−2⋅bb2\frac{15}{3} \cdot \frac{a^{3-1}}{a^{1-2}} \cdot \frac{b}{b^2}.

Simplifying the Expression

Now, let's simplify the expression by applying the rules of exponents and division.

  • 153=5\frac{15}{3} = 5
  • a3−1a1−2=a2a−1=a2−(−1)=a3\frac{a^{3-1}}{a^{1-2}} = \frac{a^2}{a^{-1}} = a^{2-(-1)} = a^3
  • bb2=1b\frac{b}{b^2} = \frac{1}{b}

Combining the Terms

Now, let's combine the terms to simplify the expression.

15a3b3ab2=5⋅a3⋅1b=5a3b\frac{15 a^3 b}{3 a b^2} = 5 \cdot a^3 \cdot \frac{1}{b} = \frac{5 a^3}{b}

Conclusion

In conclusion, we have simplified the given expression (15a3b)÷(3ab2)\left(15 a^3 b\right) \div \left(3 a b^2\right) using the rules of exponents and division. The simplified expression is 5a3b\frac{5 a^3}{b}. This expression is easier to work with and can be used to solve problems involving algebraic expressions.

Frequently Asked Questions

  • What is the rule for dividing algebraic expressions with the same base?
    • When dividing two algebraic expressions with the same base, we subtract the exponents.
  • How do we simplify an expression with exponents?
    • We apply the rules of exponents and division to simplify the expression.
  • What is the final simplified expression?
    • The final simplified expression is 5a3b\frac{5 a^3}{b}.

Final Answer

The final answer is 5a3b\boxed{\frac{5 a^3}{b}}.

Step-by-Step Solution

Here's a step-by-step solution to simplify the expression:

  1. Rewrite the expression as 15a3b3ab2\frac{15 a^3 b}{3 a b^2}.
  2. Apply the rules of exponents and division to simplify the expression.
  3. Simplify the expression by subtracting the exponents and dividing the terms.
  4. Combine the terms to simplify the expression.
  5. The final simplified expression is 5a3b\frac{5 a^3}{b}.

Tips and Tricks

  • When simplifying expressions, always apply the rules of exponents and division.
  • Use the correct order of operations to simplify the expression.
  • Combine the terms to simplify the expression.

Common Mistakes

  • Not applying the rules of exponents and division correctly.
  • Not combining the terms to simplify the expression.
  • Not using the correct order of operations.

Real-World Applications

Simplifying expressions is a crucial skill that has many real-world applications. Here are a few examples:

  • Science and Engineering: Simplifying expressions is essential in science and engineering to solve problems involving algebraic expressions.
  • Computer Programming: Simplifying expressions is a crucial skill in computer programming to write efficient code.
  • Finance: Simplifying expressions is essential in finance to calculate interest rates and investment returns.

Conclusion

In conclusion, simplifying expressions is a crucial skill that has many real-world applications. By applying the rules of exponents and division, we can simplify expressions and make them easier to work with. The final simplified expression is 5a3b\frac{5 a^3}{b}. This expression can be used to solve problems involving algebraic expressions.

Introduction

In our previous article, we simplified the expression (15a3b)÷(3ab2)\left(15 a^3 b\right) \div \left(3 a b^2\right) using the rules of exponents and division. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the rule for dividing algebraic expressions with the same base?

A: When dividing two algebraic expressions with the same base, we subtract the exponents.

Q: How do we simplify an expression with exponents?

A: We apply the rules of exponents and division to simplify the expression.

Q: What is the final simplified expression?

A: The final simplified expression is 5a3b\frac{5 a^3}{b}.

Q: What is the order of operations when simplifying expressions?

A: The order of operations is:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division (from left to right)
  4. Addition and Subtraction (from left to right)

Q: How do we handle negative exponents when simplifying expressions?

A: When simplifying expressions with negative exponents, we can rewrite the expression with positive exponents by taking the reciprocal of the expression.

Q: Can we simplify expressions with variables in the denominator?

A: Yes, we can simplify expressions with variables in the denominator by applying the rules of exponents and division.

Q: How do we simplify expressions with fractions in the numerator and denominator?

A: We can simplify expressions with fractions in the numerator and denominator by finding the least common multiple (LCM) of the denominators and multiplying both the numerator and denominator by the LCM.

Q: What is the difference between simplifying expressions and solving equations?

A: Simplifying expressions involves reducing the complexity of an expression by applying the rules of exponents and division, while solving equations involves finding the value of a variable that makes the equation true.

Q: Can we simplify expressions with multiple variables?

A: Yes, we can simplify expressions with multiple variables by applying the rules of exponents and division.

Q: How do we simplify expressions with absolute values?

A: We can simplify expressions with absolute values by removing the absolute value sign and considering both positive and negative cases.

Q: Can we simplify expressions with radicals?

A: Yes, we can simplify expressions with radicals by applying the rules of radicals and exponents.

Tips and Tricks

  • When simplifying expressions, always apply the rules of exponents and division.
  • Use the correct order of operations to simplify the expression.
  • Combine the terms to simplify the expression.
  • Be careful when handling negative exponents and variables in the denominator.
  • Use the least common multiple (LCM) to simplify expressions with fractions in the numerator and denominator.

Common Mistakes

  • Not applying the rules of exponents and division correctly.
  • Not combining the terms to simplify the expression.
  • Not using the correct order of operations.
  • Not handling negative exponents and variables in the denominator correctly.
  • Not using the least common multiple (LCM) to simplify expressions with fractions in the numerator and denominator.

Real-World Applications

Simplifying expressions is a crucial skill that has many real-world applications. Here are a few examples:

  • Science and Engineering: Simplifying expressions is essential in science and engineering to solve problems involving algebraic expressions.
  • Computer Programming: Simplifying expressions is a crucial skill in computer programming to write efficient code.
  • Finance: Simplifying expressions is essential in finance to calculate interest rates and investment returns.

Conclusion

In conclusion, simplifying expressions is a crucial skill that has many real-world applications. By applying the rules of exponents and division, we can simplify expressions and make them easier to work with. The final simplified expression is 5a3b\frac{5 a^3}{b}. This expression can be used to solve problems involving algebraic expressions.

Final Answer

The final answer is 5a3b\boxed{\frac{5 a^3}{b}}.

Step-by-Step Solution

Here's a step-by-step solution to simplify the expression:

  1. Rewrite the expression as 15a3b3ab2\frac{15 a^3 b}{3 a b^2}.
  2. Apply the rules of exponents and division to simplify the expression.
  3. Simplify the expression by subtracting the exponents and dividing the terms.
  4. Combine the terms to simplify the expression.
  5. The final simplified expression is 5a3b\frac{5 a^3}{b}.

Frequently Asked Questions

  • What is the rule for dividing algebraic expressions with the same base?
    • When dividing two algebraic expressions with the same base, we subtract the exponents.
  • How do we simplify an expression with exponents?
    • We apply the rules of exponents and division to simplify the expression.
  • What is the final simplified expression?
    • The final simplified expression is 5a3b\frac{5 a^3}{b}.

Final Tips

  • Always apply the rules of exponents and division when simplifying expressions.
  • Use the correct order of operations to simplify the expression.
  • Combine the terms to simplify the expression.
  • Be careful when handling negative exponents and variables in the denominator.
  • Use the least common multiple (LCM) to simplify expressions with fractions in the numerator and denominator.