# What are the 7 laws of exponents?

Exponents, often known as powers, are numbers that indicate how many times a base number may be multiplied by itself. For instance, the number 43 instructs you to multiply four by itself three times. The base is the number being raised by a power, whereas the exponent or power is the superscript number above it.

For example, 5^{3} = 5 × 5 × 5 = 125; the equation is written as “five to the power of three.” The power of two is also known as “squared,” whereas the power of three is known as “cubed.” When calculating the area or volume of various forms, these words are frequently employed.

### Laws of Exponents

You must understand seven exponent rules, often known as exponent laws. Each rule demonstrates how to answer various sorts of arithmetic problems as well as how to multiply, divide, and add exponents.

- Product of powers rule
- Quotient of powers rule
- Power of a power rule
- Power of a product rule
- Power of a quotient rule
- Zero power rule
- Negative exponent rule

**Product of Powers Rule**

Keep the bases the same when multiplying two bases of the same value, and then add the exponents together to get the result.

5^{2} × 5^{3} =?

Keep the base values the same because they’re both five, and then add the exponents together (2+3).

5^{2} × 5^{3} = 5^{5}

To get the answer, multiply five by itself five times.

5^{5} = 5 × 5 × 5 × 5 × 5 = 3125

**Quotient of Powers Rule**

Multiplication and division are diametrically opposed; similarly, the quotient rule is the polar opposite of the product rule. Keep the base constant when dividing two bases with the same value, and then subtract the exponent values.

4^{5} ÷ 4^{3} =?

Because both bases in this equation are four, they remain the same. Then subtract the divisor from the dividend using the exponents.

4^{5} ÷ 4^{3} =4^{2}

Finally, if necessary, simplify the equation.

4^{2} = 4 × 4 = 16

**Power of a Power Rule**

This rule explains how to solve equations in which one power is boosted by another.

(2^{4})^{4} =?

Multiply the exponents together in equations like the one above while keeping the base constant.

(2^{4})^{4} = 2^{16}

**Power of a Product Rule**

Distribute the exponent to each portion of the base when multiplying any base by an exponent.

(x × y)^{3} =?

The power of three must be allocated to both the x and y variables in this equation.

(x × y)^{3} = x^{3} y^{3}

If there are exponents linked to the base, this rule also applies.

(x^{2 }× y^{2})^{3} = x^{6} y^{6}

**Power of a Quotient Rule**

Simply said, a quotient is a result of dividing two numbers. You’re boosting a quotient by power in this rule. The exponent, like the power of a product rule, must be spread to all values within the brackets to which it is connected.

(2/3)^{4} =?

In this case, multiply both variables within the brackets by four.

(2/3)^{4} = 2^{4}/3^{4}

**Zero Power Rule**

Any base that has been raised to the power of zero equals one.

^{ }3^{0} = 1

The quotient of powers rule is the simplest method to describe this concept.

3^{3}/3^{3} =?

Subtract the exponents from each other using the quotient of powers rule, which cancels them out and leaves only the base. Any integer is equal to one when split by itself.

3^{3}/3^{3} = 3/3 = 1

**Negative Exponent Rule**

When a negative exponent is used to raise a number, convert it to a reciprocal to make the exponent positive. To make the base negative, don’t use the negative exponent. Reciprocals are the numbers that are multiplied by one to produce the value of one. For instance, multiply two by 12 to make one.

x^{-2} =?

To reciprocate a number, use the following formula:

- Make a fraction out of the number (put it over one)
- Change the denominator to the numerator and vice versa.
- When a negative number in a fraction moves to the right, it becomes a positive number.

x^{-2} = 1/x^{2}

### Table: Laws of Exponents

Formula Name | Formula |
---|---|

Product of Powers Rule | x^{m }× x^{n} = x^{m+n} |

Quotient of Powers Rule | x^{m} ÷ x^{n} = x^{m-n} |

Power of a Power Rule | (x^{m})^{n} = x^{m×n} |

Power of Product Rule | (x × y)^{n} = x^{n} × y^{n} |

Power of Quotient Rule | (x / y)^{n} = x^{n} / y^{n} |

Zero Power Rule | x^{0} =1 |

Negative Exponent Rule | x^{-n} = (1/x^{n}) |

### Sample Questions

**Question 1: What is the simplification of 7 ^{3} × 7^{1}?**

**Solution:**

7

^{3}× 7^{1}= 7^{3+1}= 7^{4}

**Question 2: Simplify and find the value of 10 ^{2}/5^{2}.**

**Solution:**

We can write the given expression as;

10

^{2}/5^{2}= (10/5)^{2}= 2^{2}= 4

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