Simplify each of the following i ii iii iv

Selina Solutions Class 9 Mathematics Solutions for Exercise 1(B) in Chapter 1 - Chapter 1- Rational and Irrational Numbers

Question 20 Exercise 1(B)

Simplify each of the following:

(i) $\sqrt[5]{16} \times \sqrt[5]{2}$

(ii) $\frac{\sqrt[4]{243}}{\sqrt[4]{3}}$

(iii) (𝟑 + √𝟐)(𝟒 + √𝟕)

(iv) (√𝟑 − √𝟐)²

Answer:

(i) $\begin{array}{l} \sqrt[5]{16} \times \sqrt[5]{2} \\ =16^{\frac{1}{5}} \times 2^{\frac{1}{5}} \\ =2^{4 \cdot \frac{1}{5}} \times 2^{\frac{1}{5}} \\ =2^{\frac{4}{5}} \times 2^{\frac{1}{5}} \\ =2^{\frac{4}{5}+\frac{1}{5}} \\ =2^{\frac{5}{5}} \\ =2^{1} \\ =2 \end{array}$

(ii) $\begin{aligned} & \frac{\sqrt[4]{243}}{\sqrt[4]{3}} \\ &=\frac{\sqrt[4]{3^{5}}}{\sqrt[4]{3}} \\ &=\frac{3^{5 \frac{1}{4}}}{3^{\frac{1}{4}}} \\ &=\frac{3^{\frac{5}{4}}}{3^{\frac{1}{4}}} \\ &=3^{\frac{5}{4}-\frac{1}{4}} \\ &=3^{\frac{4}{4}} \\ &=3 \end{aligned}$

(iii) $\begin{array}{l} (3+\sqrt{2})(4+\sqrt{7}) \\ =3 \times 4+3 \times \sqrt{7}+4 \times \sqrt{2}+\sqrt{2} \times \sqrt{7} \\ =12+3 \sqrt{7}+4 \sqrt{2}+\sqrt{14} \end{array}$

(iv) $\begin{array}{l} (\sqrt{3}-\sqrt{2})^{2} \\ =(\sqrt{3})^{2}+(\sqrt{2})^{2}-2 \times \sqrt{3} \times \sqrt{2} \\ =3+2-2 \sqrt{6} \\ =5-2 \sqrt{6} \end{array}$

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