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Exercise 2.3 (Solutions)

Question 1

Write each radical expression in exponential notation and each exponential expression in radical notation, Do not simplify.

  • (i) $\sqrt[3]{-64}$ *(ii) $2^{35}$
  • (iii) $-7^\frac{1}{3}$ * (iv) $y^\frac{-2}{3}$

Solution

* (i) $\sqrt[3]{-64} = -64^\frac{1}{3}$ ( Exponential form)

* (ii) $2^\frac{3}{5} = \sqrt[5]{2}^{3}$ (Radical form)

* (iii) $-7^\frac{1}{3} = -\sqrt[3]{7}$ (Radical form)

* (iv) $y^\frac{-2}{3} = \sqrt[3]{y}^{-2}$ (Radical form)

Question 2

Tell whether the following statements are true or false
* (i) $ 5^\frac{1}{5} = \sqrt{5}$
* (ii) $2^\frac{2}{3} = \sqrt[3]{4}$
* (iii) $\sqrt{49} = \sqrt{7}$
* (iv) $\sqrt[3]{x}^{27} = x^3$

Solution

  • (i) False
  • (ii) True
  • (iii) False
  • (iv) False

Question 3

Simplify the following radical expression

  • (i) $\sqrt[3]{-125}$
  • (ii) $\sqrt[4]{32}$
  • (iii) $\sqrt[5]{\frac{3}{32}}$
  • (iv) $\sqrt[3]{\frac{-8}{27}}$

Soluton

(i) $$\begin{array}{cl} \sqrt[3]{-125} &= \sqrt[3]{-5^3}\\ & = {-5}^{3\times\frac{1}{3}}\\ &= {-5} \end{array}$$

(ii) $$\begin{array}{cl} \sqrt[4]{32} &= \sqrt[4]{{2}^5}\\ &= \left(2^{4}\times{2}\right)^\frac{1}{4}\\ &= \left(2^{4})^\frac{1}{4}\right)\times\sqrt[4]{2}\\ &= 2\sqrt[4]{2} \end{array}$$

(iii) $$\begin{array}{cl} \sqrt[5]{\frac{3}{32}} &= \left(\frac{3}{{2^5}}\right)^\frac{1}{5}\\ &= \frac{\sqrt[5]{3}}{2} \end{array}$$

(iv) $$\begin{array}{cl} \sqrt[3]{\frac{-8}{27}} &= \sqrt[3]{\left(\frac{-2^3}{3^3}\right)}\\ &= \left(\frac{-2}{3}\right)^{3\times\frac{1}{3}}\\ &= \frac{-2}{3} \end{array}$$



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