The exponent rules are the laws used to simplify expressions with exponents. Many arithmetic operations such as addition, subtraction, multiplication, and division can be conveniently performed in quick steps using the laws of exponents. These rules also help simplify numbers with complex powers that involve fractions, decimals, and roots.

Let's learn more about the different exponent rules involving different types of base and exponent numbers.

1. | What are exponential rules? |

2. | Laws of Exponents |

3. | Law of product of exponents |

4. | Law of the quotient of exponents |

5. | Zero exponent set |

6. | Negatives Exponentengesetz |

7. | Power of a power Law of exponents |

8. | Power of the product exponent rule |

9. | Power of a quotient rule of exponents |

10 | Bruche exponent rule |

11 | Table of Exponent Rules |

12 | Frequently asked questions about exponent rules |

## What are exponential rules?

**exponent rules**, also known as "exponent laws" or "exponent properties", simplify the simplification of expressions with exponents. These rules are useful for simplifying expressions that havedecimals,fractions,irrational numbers, jnegative integersas its exponents.

For example, if we have to solve 3^{4}× 3^{2}, we can easily do this using one of the exponent rules that says a^{Metro}× and^{Norte}= and^{Meter + Norden}. With this rule, we just add the exponents to get the result while keeping the base the same, i.e. 3^{4}× 3^{2}= 3^{4 + 2}= 3^{6}. Likewise, expressions with higher exponent values can be easily solved using the exponent rules. Here is the list of exponent rules.

- A
^{0}= 1 - A
^{1}= and - A
^{Metro}× and^{Norte}= and^{m+n} - A
^{Metro}/ A^{Norte}= and^{m-n} - A
^{-m}= 1/a^{Metro} - (A
^{Metro})^{Norte}= and^{Minnesota} - (ab)
^{Metro}= and^{Metro}B^{Metro} - (a/b)
^{Metro}= and^{Metro}/B^{Metro}

## Laws of Exponents

The various exponent rules are also referred to as**Laws of Exponents**(Ö)**Properties of exponents**. The exponent laws have already been mentioned in the previous section. Most of them have specific names like product rule of exponents, quotient rule of exponents, zero rule of exponents, negative rule of exponents, etc.

Now let's learn each of them in detail.

## Law of product of exponents

IsProduct rule of exponentsused to multiply expressions with the same bases. This rule is: "To multiply two expressions with the same base, add the exponents, keeping the base the same." This rule is to add exponents with the same base. Here the rule is useful to simplify two expressions with a multiplication operation between them.

Look at the example below.

Application of the product rule of exponents | without using the rule |
---|---|

2^{3}× 2^{5}= 2^{(3 + 5)}= 2^{8} | 2^{3}× 2^{5}= (2 × 2 × 2) × (2 × 2 × 2 × 2 × 2) = 2^{8} |

This shows that without using the law, the expression involves more calculation.

## Law of the quotient of exponents

IsLaw of the quotient of the exponentsused to divide expressions with the same bases. This rule states: "To divide two expressions with the same base, subtract the exponents that keep the same base." This is useful for solving an expression without actually doing the division process. The only condition required is that the two expressions have the same base.

Here is an example.

Using the quotient law of exponents | without applying the law |
---|---|

2^{5}/2^{3}= 2^{5 - 3}= 2^{2} | 2^{5}/2^{3}= (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 ) = 2^{2} |

We can clearly see that the expression without using the law implies more calculation.

## Zero exponent set

The zeroth exponent set applies when the exponent of an expression is 0. This rule states, "Any number (other than 0) raised to 0 is 1." Note that 0^{0}is not defined, it is aindefinite form. This will help us understand that the value of a zero exponent is always equal to 1, regardless of the base.

Here is an example.

Using the zeroth exponent law | without applying the law |
---|---|

2^{0}= 1 | 2^{0}= 2^{5 - 5}^{ }= 2^{5}/2^{5}= (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2 × 2) = 1 |

Using the law we simply get 2^{0}= 1. Alternatively, without using the law, we can understand the same law with a larger number of steps.

## Negatives Exponentengesetz

The negative exponent law is used when an exponent is a negative number. This rule states: “To convert a negative exponent to a positive exponent, themutuallyshould be taken". The expression is transferred from thecountertodenominatorwith the sign change of the values of the exponents.

Here is an example.

Using the negative exponent law | without applying the law |
---|---|

2^{-2}= 1/(2^{2}) | 2^{-2}= 2^{0-2}= 2^{0}/2^{2}= 1/(2)^{2} |

With the law we can solve it at once, like 2^{-2}= 1/(2^{2}). Alternatively, without application of the law, the process is lengthy.

## Power of a power Law of exponents

Is 'Power of a power law of exponents' is used to express expressions of the form (a^{Metro})^{Norte}. This rule says, "If we have a single base with two exponents, just multiply the exponents." The two exponents are available one above the other. These can be conveniently multiplied to a single exponent.

Here is an example.

Using the power of a power law of exponents | without applying the law |
---|---|

(2^{2})^{3}= 2^{6} | (2^{2})^{3}=(2^{2})(2^{2})(2^{2}) = (2 · 2) (2 · 2) (2 · 2) = 2^{6} |

Using this law reduces the calculation process.

## Power of the product exponent rule

The 'rule of the power of a product of exponents' is used to find the result of a product raised to an exponent. This law states: "Distribute the exponent to each multiplicand of the product."

Here is an example.

Application of the power product of the exponent rule | without using the rule |
---|---|

(xy)^{3}= x^{3}.y^{3} | (xy)^{3}=(xy).(xy).(xy) = (x.x.x).(y.y.y) x^{3}.y^{3} |

With the law, (xy)^{3}= x^{3}.y^{3}. On the other hand, the same thing can be expressed in several steps without applying the law. (xy)^{3}=(xy).(xy).(xy) = (x.x.x).(y.y.y) x^{3}.y^{3}

## Power of a quotient rule of exponents

The rule of power of a quotient of exponents is used to find the result of a quotient raised to an exponent. This law states: "Distribute the exponent in both the numerator and the denominator." Here the bases are different and the exponents are the same for both bases.

Here is an example of the exponent rule given above.

Use the power of a quotient-of-exponent rule | without using the rule |
---|---|

(x/y)^{3}= x^{3}/Year^{3} | (x/y)^{3}= x/y. x/y. x/y = x^{3}/Year^{3} |

We can use the law and solve it simply, and we can also solve the same expression without the law, which requires several steps.

## Bruche exponent rule

IsRule of Fractional Exponentssays a^{1/n}=^{Norte}√a. That is, if we have a fractional exponent, it will result in radicals. for example a^{1/2}= √a, a^{1/3}= ∛a etc. This rule is further extended for complex fractional exponents like a^{Minnesota}. Using the rule of power of an exponent power (which we studied in one of the previous sections)

A^{Minnesota}= (and^{Metro})^{1/n}

Using the fractional exponent rule, this fractional power now becomes a radical.

A^{Minnesota}=^{Norte}√(and^{Metro})

This is also used as an alternative form of the rule for fractional exponents. Thus, this rule is defined in two ways:

- A
^{1/n}=^{Norte}√a - A
^{Minnesota}=^{Norte}√(and^{Metro})

## Table of Exponent Rules

The exponent rules explained above can be summarized in a diagram as shown below.

Name of the exponent rules | Ruler |
---|---|

Zero Exponent Rule | A^{0}= 1 |

Identity Exponent Rule | A^{1}= and |

product rule | A^{Metro}× and^{Norte}= and^{m+n} |

quotient rule | A^{Metro}/A^{Norte}= and^{Minnesota} |

Negative Exponentenregel | A^{-Metro}= 1/a^{Metro}; (a/b)^{-Metro}= (b/a)^{Metro} |

rule of power of a power | (A^{Metro})^{Norte}= and^{Minnesota} |

rule of power of a product | (ab)^{Metro}= and^{Metro}B^{Metro} |

Rule of the power of a quotient | (a/b)^{Metro}= and^{Metro}/B^{Metro} |

**Tips on the exponent rules:**

- If a fraction has a negative exponent, then we take the reciprocal of the fraction to make the exponent positive, i.e. (a/b)
^{-Metro}= (b/a)^{Metro} - We can convert aRadicalsinto an exponent using the following rule: a
^{1/n}=^{Norte}√a

**☛ Related items**

- exponential equations
- irrational exponents
- Exponential Rules Calculator

## Frequently asked questions about exponent rules

### What are exponential rules in mathematics?

**exponent rules**are those laws used to simplify expressions with exponents. These laws are also useful for simplifying expressions involving decimals, fractions, irrational numbers, and negative numbers.whole numbersas its exponents. For example, if we need to solve 34^{5}× 34^{7}, we can use the exponent rule, which says a^{Metro}× and^{Norte}= and^{m+n}, also 34^{5}× 34^{7}= 34^{5 + 7}= 34^{12}. Some rules for exponents are listed below:

- Produktregel: a
^{Metro}× and^{Norte}= and^{m+n}; - Quotientenregel: a
^{Metro}/A^{Norte}= and^{Minnesota}; - Rule for negative exponents: a
^{-Metro}= 1/a^{Metro}; - Rule of the power of a power: (a
^{Metro})^{Norte}= and^{Minnesota}.

### What are the 8 exponent laws?

The 8 sets of exponents can be listed as follows:

- Law of the zero exponent: a
^{0}= 1 - Law of the Exponent of Identity: a
^{1}= and - Product law: a
^{Metro}× and^{Norte}= and^{m+n} - Quotientengesetz: a
^{Metro}/A^{Norte}= and^{Minnesota} - Law of negative exponents: a
^{-Metro}= 1/a^{Metro} - power of a power: (a
^{Metro})^{Norte}= and^{Minnesota} - Performance of a product: (from)
^{Metro}= and^{Metro}B^{Metro} - Power of a quotient: (a/b)
^{Metro}= and^{Metro}/B^{Metro}

### What is the purpose of the exponent rules?

The purpose of the exponent rules is to simplify exponential expressions into fewer steps. For example, without using the rules of exponents, the expression 2^{3}× 2^{5}is written as (2 × 2 × 2) × (2 × 2 × 2 × 2 × 2) = 2^{8}. With the help of the exponent rules, this can now be simplified to 2 in just two steps^{3}× 2^{5}= 2^{(3 + 5)}= 2^{8}.

### How do you prove the exponent laws?

The laws of exponents can be easily proved by expanding the terms. the exponentialExpressionexpands by writing the base as many times as the value of the power. The exponent of the form a^{Norte}is written as a × a × a × a × a × .... n times. By multiplying we can also get the final value of the exponent. For example, let's solve 4^{2}× 4^{4}. Using the 'product law' of exponents, which states that a^{Metro}× and^{Norte}= and^{m+n}, we get 4^{2}× 4^{4}= 4^{2 + 4}= 4^{6}. This can be expanded and verified as (4 × 4) × (4 × 4 × 4 × 4) = 4096. We know that the value of 4^{6}is also 4096. Therefore, the exponent rules can be proved by extendingConditions.

### What are the rules for exponents when the bases are the same?

If the bases are the same then all the laws of exponents can be applied. For example to solve 3^{12}÷ 3^{4}, we can do that 'QuotientRule' of the exponents in which theexponentsYou remain. So, 3^{12}÷ 3^{4}will be 3^{12-4}= 3^{8}. Similarly to solve 4^{9}× 4^{4}, we apply the 'product rule' of exponents, in which the exponents are added. This results in 4^{9+4}= 4^{13}.

### What are the rules for exponents when the bases are different?

If the bases and powers are different, each term is solved separately and then we continue with the calculation below. For example, we add 4^{2}+ 2^{5}= (4 × 4) + (2 × 2 × 2 × 2 × 2) = 16 + 32 = 48. This process is applicable toadditive,Subtraction,multiplication, jdivision. In another example, when expressions with different bases and different powers are multiplied, each term is evaluated separately and then multiplied. For example 10^{3}× 6^{2}= 1000 × 36 = 36000.

### What is the rule for zero exponents?

The rule of zero exponents is a^{0}= 1. Here 'a' which is the base can be any number other than 0. This law states: "Any number (other than 0) raised to 0 is 1." For example 5^{0}= 1, x^{0}= 1 and 23^{0}= 1. Note, however, that 0^{0}It's undefined.

### What is the difference between exponents and powers?

Exponents and powers are sometimes called the same. But generally in power a^{Metro}, 'm' is known as an exponent. You can understand the differences in depth by clickingHere.

### Can the exponent be a fraction?

Yes, the value of the exponent can be a fraction. The exponent rule relating to the value of the exponent of the fraction is (a^{Metro})^{1/n}= and^{Minnesota}. This rule is sometimes useful for simplifying and transforming ataubin an exponent.