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Quantitative Aptitude :: Surds and Indices

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Surds and Indices Important Formulas

1. Laws of Indices:

$$\mathbb{i.}$$ $$a^{m} \times a^{n}$$ = $$a^{m + n}$$

$$\mathbb{ii.}$$ $$\frac{a^{m}}{a^{n}}$$ = $$a^{m - n}$$

$$\mathbb{iii.}$$ $$\left(a^{m}\right)^{n}$$ = $$a^{mn}$$

$$\mathbb{iv.}$$ $$\left(ab\right)^{n}$$ = $$a^{n}b^{n}$$

$$\mathbb{v.}$$ $$\left(\frac{a}{b}\right)^{n}$$ = $$\frac{a^{n}}{b^{n}}$$

$$\mathbb{vi.}$$ $$a^{0}$$ = 1

2. Surds:

Let a be rational number and n be a positive integer such that a(1/n) = a

Then, $$\sqrt[n]{a}$$ is called a surd of order n.

3. Laws of Surds:

$$\mathbb{i.}$$ $$\sqrt[n]{a}$$ = $$a^{\frac{1}{n}}$$

$$\mathbb{ii.}$$ $$\sqrt[n]{ab}$$ = $$\sqrt[n]{a} \times \sqrt[n]{b}$$

$$\mathbb{iii.}$$ $$\sqrt[n]{\frac{a}{b}}$$ = $$\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$$

$$\mathbb{iv.}$$ $$\left(\sqrt[n]{a}\right)^{n}$$ = a

$$\mathbb{v.}$$ $$\sqrt[m]{\sqrt[n]{a}}$$ = $$\sqrt[mn]{a}$$

$$\mathbb{vi.}$$ $$\left(\sqrt[n]{a}\right)^{m}$$ = $$\sqrt[n]{a^{m}}$$