# PROPERTIES OF EXPONENTS

In this section, we will learn the properties of exponents. We have listed the properties of exponents.

x^{m} ⋅ x^{n} = x^{(m + n)} | If two or more terms are multiplying with the same base, we may write only one base and add the powers. |

x= ^{m} / x^{n} x ^{(m – n)} | If two terms are dividing with the same base, we may write only one base and subtract the powers. |

(x ^{m})^{n} = x^{mn} | If we have power raised to another power, then we should multiply both the powers |

x ^{-m} = 1/x^{m} (or) (a/b) ^{-m} = (b/a)^{m} | If we have negative exponent, then we can write the reciprocal form of the given number to make the negative exponent as positive. |

x^{0} = 1 | The value of anything to the power will be 1 |

## Example problems on properties of exponents

**Example 1 :**

Simplify

2m^{2} ⋅ 3m^{3}

**Solution :**

2m^{2} ⋅ 3m^{3} = 6m^{(2 + 3)}

2m^{2} ⋅ 3m^{3} = 6m^{5}

**Example 2 :**

Simplify

**Solution :**

(2x^{3}/y^{3}) ⋅ (x^{4}/ y^{2}) = 2x^{(3 + 4)} / y^{(3 + 2)}

= 2x^{7} / y^{5}

**Example 3 :**

Simplify

**Solution :**

= 4g^{-4} b^{4} ⋅ 2gb^{-5}

= 8g^{(-4 + 1)} b^{(4 – 5)}

= 8g^{-3} b^{– 1}

= 8/g^{3}b

**Example 4 :**

Simplify

**Solution :**

For all the terms, we have same base. So, we can write only one base and add the powers.

= (6/7)^{(4 + 2 + 3)}

= (6/7)^{9}