Exercise 4.4: Introduction to Algebraic Expressions

Here is the Introduction to Exercise 4.4: Rationalization in Punjab Textbook for Class 9th Math:

Exercise 4.4:In mathematics, rationalization is the process of making a denominator of a fraction rational, i.e., to make it free of radicals.

This can be done by multiplying the numerator and denominator of the fraction by the conjugate of the denominator.

The conjugate of a number is another number with the same absolute value but with the opposite sign of the imaginary part. For example, the conjugate of 5 + 3i is 5 – 3i.

To rationalize a denominator, we can use the following steps:
Find the conjugate of the denominator.
Multiply the numerator and denominator of the fraction by the conjugate.
Simplify the expression.

Here are some examples of rationalization:

Example 1: Rationalize the denominator of the fraction 1/(2 + i).

The conjugate of 2 + i is 2 – i. So, we can multiply the numerator and denominator of the fraction by 2 – i.

1/(2 + i) * (2 – i)/(2 – i) = (2 – i)/(2^2 – (i)^2) = (2 – i)/(4 + 1) = 1/5

Example 2: Rationalize the denominator of the fraction 1/(3 – 2i).

The conjugate of 3 – 2i is 3 + 2i. So, we can multiply the numerator and denominator of the fraction by 3 + 2i.

1/(3 – 2i) * (3 + 2i)/(3 + 2i) = (3 + 2i)/(3^2 – (-2i)^2) = (3 + 2i)/(9 + 4) = 1/7

Exercise 4.4: Introduction to Algebraic Expressions. Algebraic Expressions and Algebraic Formulas, Exercise 4.4. Unit 4, Exercise 4.4 solution,

Here are the details of Exercise 4.4: Rationalization in the Algebraic Expressions chapter of the Class 9th Science textbook of Punjab:

In mathematics, rationalization is the process of making a denominator of a fraction rational, i.e., to make it free of radicals. This can be done by multiplying the numerator and denominator of the fraction by the conjugate of the denominator.

The conjugate of a number is another number with the same absolute value but with the opposite sign of the imaginary part. For example, the conjugate of 5 + 3i is 5 – 3i.

To rationalize a denominator, we can use the following steps:
Find the conjugate of the denominator.
Multiply the numerator and denominator of the fraction by the conjugate.
Simplify the expression.
Examples. Exercise 4.4:

Rationalize the denominator of the fraction 1/(2 + i).

The conjugate of 2 + i is 2 – i. So, we can multiply the numerator and denominator of the fraction by 2 – i.

1/(2 + i) * (2 – i)/(2 – i) = (2 – i)/(2^2 – (i)^2) = (2 – i)/(4 + 1) = 1/5

Rationalize the denominator of the fraction 1/(3 – 2i).

The conjugate of 3 – 2i is 3 + 2i. So, we can multiply the numerator and denominator of the fraction by 3 + 2i.

1/(3 – 2i) * (3 + 2i)/(3 + 2i) = (3 + 2i)/(3^2 – (-2i)^2) = (3 + 2i)/(9 + 4) = 1/7

Exercise 4.4

Rationalize the denominator of the following fractions:
- 1/(3 + 2i)
- 1/(5 – 3i)
- 1/(2 – 4i)
Simplify the following expressions:
- (3 + 2i)/(5 – 3i)
- (2 + 3i)/(5 + 2i)
- (4 + 5i)/(2 – 3i)

Answers

The answers to the first part of the exercise are as follows:
- 1/(3 + 2i) * (3 – 2i)/(3 – 2i) = 1/5
- 1/(5 – 3i) * (5 + 3i)/(5 + 3i) = 1/7
- 1/(2 – 4i) * (2 + 4i)/(2 + 4i) = 1/9
The answers to the second part of the exercise are as follows:
- (3 + 2i)/(5 – 3i) = (3 + 2i)(5 + 3i)/(5^2 – (-3i)^2) = 23/34
- (2 + 3i)/(5 + 2i) = (2 + 3i)(5 – 2i)/(5^2 – (2i)^2) = 11/29
- (4 + 5i)/(2 – 3i) = (4 + 5i)(2 + 3i)/(2^2 – (-3i)^2) = 17/9

Questions & Answers

Here are some questions and answers about Algebraic Expressions Exercise 4.4 Rationalization in the Class 9th Science textbook of Punjab:

Question 1: Rationalize the denominator of the fraction 1/(2 + i).

Answer:

The conjugate of 2 + i is 2 – i. So, we can multiply the numerator and denominator of the fraction by 2 – i.

1/(2 + i) * (2 – i)/(2 – i) = (2 – i)/(2^2 – (i)^2) = (2 – i)/(4 + 1) = 1/5

Question 2: Simplify the expression (3 + 2i)/(5 – 3i).

Answer:

We can simplify this expression by multiplying the numerator and denominator by the conjugate of the denominator, which is 5 + 3i.

(3 + 2i)/(5 – 3i) * (5 + 3i)/(5 + 3i) = (3 + 2i)(5 + 3i)/(5^2 – (-3i)^2) = (3 + 2i)(5 + 3i)/(25 + 9) = 23/34

Question 3: Write the steps involved in rationalizing the denominator of a fraction.

Answer:

The steps involved in rationalizing the denominator of a fraction are as follows:

Find the conjugate of the denominator.
Multiply the numerator and denominator of the fraction by the conjugate.
Simplify the expression.Exercise 4.4:

Question 4: What is the difference between rationalization and simplification?

Answer:

Rationalization is the process of making a denominator of a fraction rational, i.e., to make it free of radicals. Simplification is the process of reducing an expression to its most basic form.

In other words, rationalization is the process of removing radicals from the denominator of a fraction, while simplification is the process of reducing the number of terms in an expression without changing its value.

Question 5: Give some examples of rationalization.

Answer:

Some examples of rationalization are:

Rationalizing the denominator of the fraction 1/(2 + i) by multiplying the numerator and denominator by 2 – i.
Rationalizing the denominator of the fraction (3 + 2i)/(5 – 3i) by multiplying the numerator and denominator by the conjugate of the denominator, which is 5 + 3i.
Rationalizing the denominator of the fraction (4 + 5i)/(2 – 3i) by multiplying the numerator and denominator by the conjugate of the denominator, which is 2 + 3i.

Conclusion

Here is the conclusion about Algebraic Expressions Exercise 4.4: Rationalization in the Class 9th Science textbook of Punjab:

Rationalization is a process of making a denominator of a fraction rational, i.e., to make it free of radicals. This can be done by multiplying the numerator and denominator of the fraction by the conjugate of the denominator.

The conjugate of a number is another number with the same absolute value but with the opposite sign of the imaginary part. For example, the conjugate of 5 + 3i is 5 – 3i.

The exercises in this chapter help students understand the concept of rationalization and how to apply it to simplify expressions. They also help students develop their skills in algebra and complex numbers.

The exercises are graded from easy to difficult, so that students of all levels can find exercises that are challenging but not too difficult. The exercises are also accompanied by solutions, so that students can check their work and learn from their mistakes.Exercise 4.4:

Overall, the exercises in this chapter are a valuable resource for students who are learning about rationalization. They help students understand the concept, develop their skills, and practice applying it to simplify expressions.

Here are some additional points about rationalization:

Rationalization is not always necessary. Sometimes, it is easier to leave the denominator as it is.
Rationalization can sometimes lead to extraneous solutions. This means that the simplified expression may have solutions that are not valid for the original expression.
It is important to be careful when rationalizing expressions, as it is a relatively easy process to make a mistake.

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