Simplify The Following Expressions: (a) B 9 + B 7 \frac{b}{9} + \frac{b}{7} 9 B ​ + 7 B ​ (f) 3 U 4 − 2 U 5 \frac{3u}{4} - \frac{2u}{5} 4 3 U ​ − 5 2 U ​ (2a) 7 Ω + 7 Ω \frac{7}{\omega} + \frac{7}{\omega} Ω 7 ​ + Ω 7 ​ (h) 18 F − 3 3 G \frac{18}{f} - \frac{3}{3g} F 18 ​ − 3 G 3 ​

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In mathematics, simplifying algebraic expressions is a crucial skill that helps us solve equations and inequalities more efficiently. In this article, we will focus on simplifying five different algebraic expressions using various techniques such as finding common denominators, combining like terms, and canceling out common factors.

Simplifying Expression (a): b9+b7\frac{b}{9} + \frac{b}{7}

To simplify the expression b9+b7\frac{b}{9} + \frac{b}{7}, we need to find a common denominator. The least common multiple (LCM) of 9 and 7 is 63. We can rewrite each fraction with the common denominator as follows:

b9=b×79×7=7b63\frac{b}{9} = \frac{b \times 7}{9 \times 7} = \frac{7b}{63}

b7=b×97×9=9b63\frac{b}{7} = \frac{b \times 9}{7 \times 9} = \frac{9b}{63}

Now, we can add the two fractions together:

7b63+9b63=16b63\frac{7b}{63} + \frac{9b}{63} = \frac{16b}{63}

Therefore, the simplified expression is 16b63\frac{16b}{63}.

Simplifying Expression (f): 3u4−2u5\frac{3u}{4} - \frac{2u}{5}

To simplify the expression 3u4−2u5\frac{3u}{4} - \frac{2u}{5}, we need to find a common denominator. The LCM of 4 and 5 is 20. We can rewrite each fraction with the common denominator as follows:

3u4=3u×54×5=15u20\frac{3u}{4} = \frac{3u \times 5}{4 \times 5} = \frac{15u}{20}

2u5=2u×45×4=8u20\frac{2u}{5} = \frac{2u \times 4}{5 \times 4} = \frac{8u}{20}

Now, we can subtract the two fractions:

15u20−8u20=7u20\frac{15u}{20} - \frac{8u}{20} = \frac{7u}{20}

Therefore, the simplified expression is 7u20\frac{7u}{20}.

Simplifying Expression (2a): 7ω+7ω\frac{7}{\omega} + \frac{7}{\omega}

To simplify the expression 7ω+7ω\frac{7}{\omega} + \frac{7}{\omega}, we can combine the two fractions by adding their numerators:

7ω+7ω=7+7ω=14ω\frac{7}{\omega} + \frac{7}{\omega} = \frac{7 + 7}{\omega} = \frac{14}{\omega}

Therefore, the simplified expression is 14ω\frac{14}{\omega}.

Simplifying Expression (h): 18f−33g\frac{18}{f} - \frac{3}{3g}

To simplify the expression 18f−33g\frac{18}{f} - \frac{3}{3g}, we need to find a common denominator. However, we can simplify the expression by canceling out the common factor of 3 in the second fraction:

18f−33g=18f−1g\frac{18}{f} - \frac{3}{3g} = \frac{18}{f} - \frac{1}{g}

Now, we can rewrite the expression with a common denominator:

18f−1g=18gfg−ffg\frac{18}{f} - \frac{1}{g} = \frac{18g}{fg} - \frac{f}{fg}

Now, we can subtract the two fractions:

18gfg−ffg=18g−ffg\frac{18g}{fg} - \frac{f}{fg} = \frac{18g - f}{fg}

Therefore, the simplified expression is 18g−ffg\frac{18g - f}{fg}.

Conclusion

In this article, we have simplified five different algebraic expressions using various techniques such as finding common denominators, combining like terms, and canceling out common factors. By following these techniques, we can simplify complex algebraic expressions and make them easier to work with. Whether you are a student or a professional, simplifying algebraic expressions is an essential skill that can help you solve problems more efficiently and effectively.

Tips and Tricks

  • When simplifying algebraic expressions, always look for common factors or denominators.
  • Use the distributive property to expand expressions and make them easier to simplify.
  • Combine like terms to simplify expressions and make them easier to work with.
  • Cancel out common factors to simplify expressions and make them easier to work with.

Real-World Applications

Simplifying algebraic expressions has many real-world applications, including:

  • Solving equations and inequalities in physics and engineering
  • Modeling population growth and decay in biology and economics
  • Analyzing data and making predictions in statistics and data science
  • Solving optimization problems in business and finance

By mastering the techniques of simplifying algebraic expressions, you can apply them to a wide range of real-world problems and make a positive impact in your field.

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In our previous article, we explored the techniques of simplifying algebraic expressions. In this article, we will answer some common questions that students and professionals often have when it comes to simplifying algebraic expressions.

Q: What is the difference between simplifying and solving an algebraic expression?

A: Simplifying an algebraic expression involves reducing it to its simplest form, while solving an algebraic expression involves finding the value of the variable that makes the expression true. For example, simplifying the expression b9+b7\frac{b}{9} + \frac{b}{7} results in 16b63\frac{16b}{63}, while solving the equation b9+b7=2\frac{b}{9} + \frac{b}{7} = 2 involves finding the value of bb that makes the equation true.

Q: How do I know when to simplify an algebraic expression?

A: You should simplify an algebraic expression whenever possible, especially when working with complex expressions or when trying to solve equations or inequalities. Simplifying expressions can make them easier to work with and can help you identify patterns or relationships that may not be apparent in the original expression.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not finding a common denominator when adding or subtracting fractions
  • Not combining like terms when simplifying expressions
  • Not canceling out common factors when simplifying expressions
  • Not checking for errors when simplifying expressions

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, you can use the rule that a−n=1ana^{-n} = \frac{1}{a^n}. For example, the expression x−2x^{-2} can be simplified to 1x2\frac{1}{x^2}.

Q: How do I simplify expressions with radicals?

A: To simplify expressions with radicals, you can use the rule that a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}. For example, the expression 12\sqrt{12} can be simplified to 4⋅3=23\sqrt{4 \cdot 3} = 2\sqrt{3}.

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

A: Yes, you can simplify expressions with variables in the denominator. However, you need to be careful when simplifying expressions with variables in the denominator, as you may need to multiply both the numerator and the denominator by a constant or a variable to eliminate the variable in the denominator.

Q: How do I simplify expressions with multiple variables?

A: To simplify expressions with multiple variables, you can use the distributive property to expand the expression and then combine like terms. For example, the expression 2x+3y2x + 3y can be simplified to (2x+3y)(2x + 3y).

Q: Can I simplify expressions with fractions in the numerator or denominator?

A: Yes, you can simplify expressions with fractions in the numerator or denominator. However, you need to be careful when simplifying expressions with fractions in the numerator or denominator, as you may need to multiply both the numerator and the denominator by a constant or a variable to eliminate the fraction.

Q: How do I check my work whenifying algebraic expressions?

A: To check your work when simplifying algebraic expressions, you can:

  • Plug in values for the variables to see if the expression simplifies to the expected value
  • Use a calculator to evaluate the expression and see if it simplifies to the expected value
  • Check your work by simplifying the expression in a different way or by using a different method

By following these tips and techniques, you can simplify algebraic expressions with confidence and accuracy.