Perform The Indicated Operations, Then Simplify:1. { \frac{3b}{4a^2} + \frac{1}{8a} - \frac{5b 2}{6a 3}$}$2. { \frac{-5b^2 + 3b + 1}{6a^6}$}$3. { \frac{3b + 1 - 5b 2}{4a 2 + 8a - 6a^3}$} 4. \[ 4. \[ 4. \[ \frac{3a^2 + 18ab -

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and performing indicated operations and simplifying them is a crucial skill for students to master. In this article, we will explore four different algebraic expressions and perform the indicated operations to simplify them. We will use various mathematical techniques, including combining like terms, factoring, and canceling common factors.

Expression 1: Combining Fractions

The first expression we will simplify is:

3b4a2+18a−5b26a3\frac{3b}{4a^2} + \frac{1}{8a} - \frac{5b^2}{6a^3}

To simplify this expression, we need to find a common denominator. The least common multiple (LCM) of 4, 8, and 6 is 24. We can rewrite each fraction with a denominator of 24:

3b4a2=3bâ‹…64a2â‹…6=18b24a2\frac{3b}{4a^2} = \frac{3b \cdot 6}{4a^2 \cdot 6} = \frac{18b}{24a^2}

18a=1â‹…38aâ‹…3=324a\frac{1}{8a} = \frac{1 \cdot 3}{8a \cdot 3} = \frac{3}{24a}

5b26a3=5b2â‹…46a3â‹…4=20b224a3\frac{5b^2}{6a^3} = \frac{5b^2 \cdot 4}{6a^3 \cdot 4} = \frac{20b^2}{24a^3}

Now we can combine the fractions:

18b24a2+324a−20b224a3\frac{18b}{24a^2} + \frac{3}{24a} - \frac{20b^2}{24a^3}

We can simplify this expression by combining like terms:

18b24a2+324a−20b224a3=18b−20b224a2+324a−20b224a3\frac{18b}{24a^2} + \frac{3}{24a} - \frac{20b^2}{24a^3} = \frac{18b - 20b^2}{24a^2} + \frac{3}{24a} - \frac{20b^2}{24a^3}

We can factor out a -20b^2 from the first two terms:

−20b2(1−b2a2)24a2+324a−20b224a3\frac{-20b^2(1 - \frac{b}{2a^2})}{24a^2} + \frac{3}{24a} - \frac{20b^2}{24a^3}

We can simplify this expression by canceling common factors:

−20b2(1−b2a2)24a2+324a−20b224a3=−5b2(1−b2a2)6a2+324a−5b26a3\frac{-20b^2(1 - \frac{b}{2a^2})}{24a^2} + \frac{3}{24a} - \frac{20b^2}{24a^3} = \frac{-5b^2(1 - \frac{b}{2a^2})}{6a^2} + \frac{3}{24a} - \frac{5b^2}{6a^3}

Expression 2: Simplifying a Rational Expression

The second expression we will simplify is:

−5b2+3b+16a6\frac{-5b^2 + 3b + 1}{6a^6}

This expression is already simplified, so we don't need to perform any further operations.

Expression 3: Combining Like Terms

The third expression we will simplify is:

3b+1−5b24a2+8a−6a3\frac{3b + 1 - 5b^2}{4a^2 + 8a - 6a^3}

To simplify this expression, we need to combine like terms in the numerator:

3b+1−5b2=−5b2+3b+13b + 1 - 5b^2 = -5b^2 + 3b + 1

We can factor out a -5b^2 from the first two terms:

−5b2+3b+1=−5b2(1−35b)+1-5b^2 + 3b + 1 = -5b^2(1 - \frac{3}{5b}) + 1

We can simplify this expression by canceling common factors:

−5b2(1−35b)+1=−5b2(1−35b)+55-5b^2(1 - \frac{3}{5b}) + 1 = -5b^2(1 - \frac{3}{5b}) + \frac{5}{5}

We can simplify this expression by combining like terms:

−5b2(1−35b)+55=−5b2(1−35b)+1-5b^2(1 - \frac{3}{5b}) + \frac{5}{5} = -5b^2(1 - \frac{3}{5b}) + 1

Expression 4: Factoring a Polynomial

The fourth expression we will simplify is:

3a2+18ab−24b24a2+8a−6a3\frac{3a^2 + 18ab - 24b^2}{4a^2 + 8a - 6a^3}

To simplify this expression, we need to factor the numerator:

3a2+18ab−24b2=3(a2+6ab−8b2)3a^2 + 18ab - 24b^2 = 3(a^2 + 6ab - 8b^2)

We can factor the numerator as a difference of squares:

3(a2+6ab−8b2)=3(a+4b)(a−2b)3(a^2 + 6ab - 8b^2) = 3(a + 4b)(a - 2b)

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)4a2+8a−6a3=3(a+4b)(a−2b)2a(2a2+4a−3a3)\frac{3(a + 4b)(a - 2b)}{4a^2 + 8a - 6a^3} = \frac{3(a + 4b)(a - 2b)}{2a(2a^2 + 4a - 3a^3)}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(2a2+4a−3a3)=3(a+4b)(a−2b)2a(a(2a+4−3a2))\frac{3(a + 4b)(a - 2b)}{2a(2a^2 + 4a - 3a^3)} = \frac{3(a + 4b)(a - 2b)}{2a(a(2a + 4 - 3a^2))}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(a(2a+4−3a2))=3(a+4b)(a−2b)2a(a(2(a+2)−3a2))\frac{3(a + 4b)(a - 2b)}{2a(a(2a + 4 - 3a^2))} = \frac{3(a + 4b)(a - 2b)}{2a(a(2(a + 2) - 3a^2))}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(a(2(a+2)−3a2))=3(a+4b)(a−2b)2a(a(2a+2−3a2))\frac{3(a + 4b)(a - 2b)}{2a(a(2(a + 2) - 3a^2))} = \frac{3(a + 4b)(a - 2b)}{2a(a(2a + 2 - 3a^2))}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(a(2a+2−3a2))=3(a+4b)(a−2b)2a(a(2(a−a2+1)−3a2))\frac{3(a + 4b)(a - 2b)}{2a(a(2a + 2 - 3a^2))} = \frac{3(a + 4b)(a - 2b)}{2a(a(2(a - a^2 + 1) - 3a^2))}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(a(2(a−a2+1)−3a2))=3(a+4b)(a−2b)2a(a(2a−a2+2−3a2))\frac{3(a + 4b)(a - 2b)}{2a(a(2(a - a^2 + 1) - 3a^2))} = \frac{3(a + 4b)(a - 2b)}{2a(a(2a - a^2 + 2 - 3a^2))}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(a(2a−a2+2−3a2))=3(a+4b)(a−2b)2a(a(2a−a2−3a2+2))\frac{3(a + 4b)(a - 2b)}{2a(a(2a - a^2 + 2 - 3a^2))} = \frac{3(a + 4b)(a - 2b)}{2a(a(2a - a^2 - 3a^2 + 2))}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(a(2a−a2−3a2+2))=3(a+4b)(a−2b)2a(a(2a−4a2+2))\frac{3(a + 4b)(a - 2b)}{2a(a(2a - a^2 - 3a^2 + 2))} = \frac{3(a + 4b)(a - 2b)}{2a(a(2a - 4a^2 + 2))}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(a(2a−4a2+2))=3(a+4b)(a−2b)2a(a(2(a−2a2+1))\frac{3(a + 4b)(a - 2b)}{2a(a(2a - 4a^2 + 2))} = \frac{3(a + 4b)(a - 2b)}{2a(a(2(a - 2a^2 + 1))}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(a(2(a−2a2+1))=3(a+4b)(a−2b)2a(a(2(a−2a2+1))\frac{3(a + 4b)(a - 2b)}{2a(a(2(a - 2a^2 + 1))} = \frac{3(a + 4b)(a - 2b)}{2a(a(2(a - 2a^2 + 1))}

We can simplify this expression by canceling common factors:

3(a+4b)(a−2b)2a(a(2(a−2a2+1))=3(a+4b)(a−2b)2a(a(2(a−2a2+1))\frac{3(a + 4b)(a - 2b)}{2a(a(2(a - 2a^2 + 1))} = \frac{3(a + 4b)(a - 2b)}{2a(a(2(a - 2a^2 + 1))}

We can simplify this expression by canceling common factors:

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify any common factors that can be canceled out. This can include factoring out a greatest common factor (GCF) from the numerator and denominator.

Q: How do I combine like terms in an algebraic expression?

A: To combine like terms in an algebraic expression, you need to identify the terms that have the same variable and exponent. You can then add or subtract these terms to simplify the expression.

Q: What is the difference between a rational expression and a polynomial expression?

A: A rational expression is an expression that contains a fraction with variables in the numerator and/or denominator. A polynomial expression, on the other hand, is an expression that contains only variables and constants, with no fractions.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to find a common denominator for the fractions in the numerator and denominator. You can then combine the fractions and simplify the resulting expression.

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12.

Q: How do I find the LCM of two algebraic expressions?

A: To find the LCM of two algebraic expressions, you need to find the LCM of the coefficients and the LCM of the variables. You can then use these values to create a common denominator for the fractions in the numerator and denominator.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change. A constant, on the other hand, is a value that does not change.

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, you need to identify any common factors that can be canceled out. You can then combine like terms and simplify the resulting expression.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to check your work and make sure that the expression is simplified as much as possible.

Common Algebraic Expression Simplification Mistakes

  • Not canceling common factors: Make sure to cancel out any common factors in the numerator and denominator.
  • Not combining like terms: Make sure to combine like terms in the numerator and denominator.
  • Not finding the LCM: Make sure to find the LCM of the coefficients and variables in the numerator and denominator.
  • Not simplifying the expression: Make sure to simplify the expression as much as possible.

Conclusion

Simplifying algebraic expressions is an important skill that can be used in a variety of mathematical applications. By following the steps outlined in this article, you can simplify even the most complex algebraic expressions. Remember to always check your work and make sure that the expression is simplified as much as possible.