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Q2(ii):
Simplify each of the following expressions:
(ii) $(3 + \sqrt{3})(3 - \sqrt{3})$
Solution :
Initial Setup & Given Expression
We are tasked with simplifying the following irrational algebraic expression:
$(3 + \sqrt{3})(3 - \sqrt{3})$
Step 1: Identification of the Algebraic Identity
The given expression is in the form of the product of the sum and difference of two binomial terms. We can optimize the simplification process by applying the fundamental algebraic identity for the difference of two squares.
[Per the algebraic identity for the difference of two squares]:
$(a + b)(a - b) = a^2 - b^2$
Step 2: Variable Mapping and Substitution
By comparing our specific expression $(3 + \sqrt{3})(3 - \sqrt{3})$ with the standard identity $(a + b)(a - b)$, we can map the variables as follows:
- $a = 3$
- $b = \sqrt{3}$
Substituting these values into the right-hand side of the identity ($a^2 - b^2$), we obtain:
$(3)^2 - (\sqrt{3})^2$
Step 3: Simplification of Individual Terms
We now evaluate the square of each term independently.
First Term ($a^2$):
$(3)^2 = 3 \times 3 = 9$
Second Term ($b^2$):
[Per the definition of a principal square root, the square of a square root of a non-negative real number $x$ returns the number itself: $(\sqrt{x})^2 = x$]
$(\sqrt{3})^2 = 3$
Step 4: Final Arithmetic Operation
Substitute the simplified values back into the expression from Step 2:
$9 - 3 = 6$
Final Solution: The simplified value of the expression $(3 + \sqrt{3})(3 - \sqrt{3})$ is 6.
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.4
- Q1(i): Classify the following numbers as rational or irrational: (i) $2 - \sqrt{5}$
- Q1(ii): Classify the following numbers as rational or irrational: (ii) $(3 + \sqrt{23}) - \sqrt{23}$
- Q1(iii): Classify the following numbers as rational or irrational: (iii) $\frac{2\sqrt{7}}{7\sqrt{7}}$
- Q1(iv): Classify the following numbers as rational or irrational: (iv) $\frac{1}{\sqrt{2}}$
- Q1(v): Classify the following numbers as rational or irrational: (v) $2\pi$
- Q2(i): Simplify each of the following expressions: (i) $(3 + \sqrt{3})(2 + \sqrt{2})$
- Q2(iii): Simplify each of the following expressions: (iii) $(\sqrt{5} + \sqrt{2})^2$
- Q2(iv): Simplify each of the following expressions: (iv) $(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})$
- Q3: Recall, $\pi$ is defined as the ratio of the circumference (say $c$) of a circle to its diameter (say $d$). That is, $\pi = \frac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction?
- Q4: Represent $\sqrt{9.3}$ on the number line.
- Q5(i): Rationalise the denominators of the following: (i) $\frac{1}{\sqrt{7}}$
- Q5(ii): Rationalise the denominators of the following: (ii) $\frac{1}{\sqrt{7} - \sqrt{6}}$
- Q5(iii): Rationalise the denominators of the following: (iii) $\frac{1}{\sqrt{5} + \sqrt{2}}$
- Q5(iv): Rationalise the denominators of the following: (iv) $\frac{1}{\sqrt{7} - 2}$
