# Potencias y raíces en inglés

## Potencias (Powers)

$3^{4}$=3.3.3.3=81

You read $3^{4}$ as:

three to the fourth power

three to the power of 4

In this case 3 is the base and 4 is the exponent.

### Especial cases

$4^{2}$ is read as:

four to the second power

four squared (the most common)

four to the power of two

four to the square power

$5^{3}$ is read as

five cubed (the most common)

five to the third power

five to the power of three

### Laws of powers

$a^{0}=1$

$a^{n}.a^{m}=a^{n+m}$

$\frac{a^{n}}{a^{m}}=a^{n-m}$

$\left&space;(&space;a^{n}&space;\right&space;)^{m}=a^{n.m}$

$a^{-n}=\frac{1}{a^{n}}$

$a^{n}.b^{n}=\left&space;(&space;a.b&space;\right&space;)^{n}$

$\left&space;(&space;\frac{a}{b}&space;\right&space;)^{n}=\frac{a^{n}}{b^{n}}$

$\left&space;(&space;\frac{a}{b}&space;\right&space;)^{-n}=\left&space;(&space;\frac{b}{a}&space;\right&space;)^{n}$

### Ejercicios de potencias en inglés

1. Calculate:

$2^{5}=$

$\left&space;(&space;\frac{3}{5}&space;\right&space;)^{2}=$

$4^{3}=$

2. Reduce the powers

a) $\frac{m^{3}.m^{2}}{m^{5}}=$

b) $\frac{6^{3}.12^{4}}{24^{2}}=$

c) $\frac{2^{-3}.4^{3}}{8^{-2}}=$

3. Express the  following numbers as a power of base ten:

a) 0.1=

b) 0.001=

c) One hundredth=

d) One millionth=

$\sqrt[n]{a}$ is called radical, n index of the root, a radicand.

The opposite of squaring a number is finding its square root.

$\sqrt{36}=6$  because  $6^{2}=36$

### Raíz cúbica (cube root)

The opposite of cubing a number is findingits cube root.

$\sqrt[3]{27}=3$  because  $3^{3}=27$

### Otras raíces (other roots)

We can also use this same pattern to calculate roots with an index number greater than 3

$\sqrt[4]{81}=3$  because  $3^{4}=81$

### Ejercicios de radicales en inglés

1. Calculate:

a) $\sqrt[4]{16}=$

b) $\sqrt{\frac{4}{25}}=$

c) $\sqrt[3]{\frac{8}{125}}=$

d) $\sqrt{2.56}=$

e) $\sqrt[3]{0.001}$=

If an exponent is equal to the index, the factor goes outside the radical.

Examples:

$\sqrt{45}=\sqrt{3^{2}.5}=3\sqrt{5}$

$\sqrt[3]{54}=\sqrt[3]{3^{3}.2}=3\sqrt[3]{2}$

### $\sqrt[np]{a^{p}}=\sqrt[n]{a}$

Examples:

$\sqrt[6]{216}=\sqrt[6]{2^{3}.3^{3}}=\sqrt{2.3}=\sqrt{6}$

$\sqrt[8]{x^{2}.y^{4}.z^{6}}=\sqrt[4]{x.y^{2}.z^{3}}$

$\left&space;(&space;\sqrt[n]{a}&space;\right&space;)^{p}=\sqrt[n]{a^{p}}$

Example:

$(\sqrt[6]{3})^{2}=\sqrt[6]{3^{2}}=\sqrt[3]{3}$

Exercises

1. Calculate and reduce:

a) $\sqrt{8}-3\sqrt{18}+5\sqrt{50}=$

b) $\sqrt[3]{3}+3\sqrt[3]{24}-\sqrt[3]{2}+\sqrt[3]{16}=$

c) $\sqrt{2}.\sqrt{12}.\sqrt{3}=$

d) $\frac{\sqrt[3]{3}.\sqrt[6]{12}}{\sqrt[4]{6}}=$

2. Simplify:

a) $\left&space;(&space;\sqrt{x^{3}}&space;\right&space;)^{2}$=

b) $\sqrt[12]{2^{6}.3^{9}.5^{3}}=$

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