**Introduction**

Exponents are a fundamental aspect of mathematics that every student will encounter. They are one of the keys to unlocking more complex mathematical ideas and operations. This article will concentrate on specific rules that make dealing with exponents easier and simpler.

**Grade Appropriateness**

This topic is typically introduced around 6th grade in most educational curricula and further reinforced and deeply understood in 7th and 8th grades. However, the understanding and application of these laws carry on throughout high school and beyond.

**Math Domain**

The laws of integer exponents fall under Number Theory and Algebra.

**Applicable Common Core Standards**

The Common Core Standards related to this topic are:

*CCSS.MATH.CONTENT.6.EE.A.1:* Write and evaluate numerical expressions involving whole-number exponents.

*CCSS.MATH.CONTENT.8.EE.A.1:* Know and apply the properties of integer exponents to generate equivalent numerical expressions.

**Definition of the Topic**

An ** exponent** is a number that indicates the number of times a base number is multiplied by itself. If n is a positive integer, then an is the product of multiplying ‘a’ by itself ‘n’ times.

The ** laws of integral exponents** govern the manipulation of expressions involving exponents.

**Key Concepts**

For any nonzero term a where m and n are real numbers, the laws of integer exponents include the following:

Product of Powers Rule When multiplying like bases, you can add the exponents: a ^{m}⋅ a^{n} = a^{m+n}. | Quotient of Powers RuleWhen dividing like bases, you can subtract the exponents: $\frac{a^m}{a^n}$ = a ^{m-n}. |

Power of a Power Rule When raising a power to a power, you can multiply the exponents: (a ^{m})^{n} = a^{m⋅n}. | Zero Exponent RuleAny nonzero number raised to the power of zero is 1: a ^{0}= 1. |

Negative Exponent RuleA nonzero number raised to a negative exponent is the reciprocal of the number raised to the opposite positive exponent: a ^{-n} =$\frac{1}{a^n}$. |

**Discussion with Illustrative Examples**

**Exponents**

Exponents are used to show the repeated multiplication of a number by itself.

Say, for example, 5×5×5×5 can be expressed as 5^{4}. The exponent is 4, meaning the number of times the base number 5 is multiplied.

**Laws of Integral Exponents**

The Laws of Integral Exponents are fundamental math concepts that guide in evaluating complex expressions, especially those involving exponents. These laws include the following:

**Product Rule**

Copy the base and add the exponents when multiplying expressions with the same base but different exponents.

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

**Examples**

1. b^{3}.b^{5}=(b⋅b⋅b)(b⋅b⋅b⋅b⋅b)=b^{8}

** **Simplifying this expression using the product rule, we have,

b^{3}.b^{5}=b^{3+5}=b^{8}

2. (c^{2}d4)(cd^{3})=(c^{2+1})(d^{4+3})=c^{3}d^{7}

Copy the base and subtract the exponents when dividing expressions with the same base.

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

**Examples**

1. $\frac{b^6}{b^4}$=$\frac{b⋅b⋅b⋅b⋅b⋅b}{b⋅b⋅b⋅b}$

When we simplify this expression using the quotient rule, we will have,

$\frac{b^6}{b^4}$=b^{6-4}=b^{2}** **

2. $\frac{c^9d^5}{c^3d^2}$=c^{9-3}d^{5-4}=c^{6}d^{3}

**Power Rule**

When an expression with an exponent is raised to another integer, multiply these exponents.

(a^{m})^{n}=a^{m⋅n}

**Examples**

1. (b^{4})^{3}=b^{4}.b^{4}.b^{4}=(b^{4}.b^{4}.b^{4}.b^{4})(b^{4}.b^{4}.b^{4}.b^{4})(b^{4}.b^{4}.b^{4}.b^{4})=b^{12}

Simplifying the expression applying the power rule, we get,

(b^{4})^{3}=b^{4×3}=b^{12}

2. (3c^{2}d^{4})^{3}=(3)^{3}(c^{2})^{3}(d^{4})^{3}=3^{3}c^{2×3}d^{4×3}=27c^{6}d^{12}

**Negative Exponent**

When the expression is raised to a negative integer, we get the reciprocal of the base number.

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

**Examples**

1. b^{-6}=$\frac{1}{b^6}$

2. $\frac{c^{-5}}{d^{-3}}$=$\frac{d^3}{c^5}$

**Zero Exponent**

Any expression raised to zero is equal to 1.

a^{0}=1

**Examples**

1. b^{0}=1

2. 10^{0}=1

3. (c^{2}d^{4})^{0}=1

**Examples with Solutions**

**Example 1**

Simplify the following expressions by applying the appropriate law/s of exponents.

i. (2x^{3}y^{2})^{4}

ii.$\frac{4a^5b^6}{8a^2b^3}$

iii. (2a^{2}+3b^{2}-c)^{0}

**Solution**

i. (2x^{3}y^{2})^{4}=(2)^{4} (x^{3})^{4}(y^{2})^{4}=2^{4}x^{3⋅4}y^{2⋅4}=16x^{12}y^{8}** ***(Power Rule and Product Rule)*

ii. $\frac{4a^5b^6}{8a^2b^3}$=$\frac{1}{2}$a^{5-2}b^{6-3}=$\frac{1}{2}$a^{3}b^{3}** ***(Quotient Rule)*

iii. (2a^{2}+3b^{2}-c)^{0}=0** ***(Zero Exponent Rule)*

**Example 2**

Simplify the following expressions.

i. 3^{-5}

ii.$\frac{a^{-4}b^{-5}c^{-1}}{a^2b^2c^2}^2$

**Solution**

i. 3^{-5}=$\frac{1}{3^5}$=$\frac{1}{243}$

ii.($\frac{a^{-4}b^{-5}c^{-1}}{a^2b^2c^2}$)^{2}=(a^{-4-2})(b^{-5-2})(c^{-1-2})=a^{-6}b^{-7}c^{-3}=$\frac{1}{a^6b^7c^3}$

**Real-life Application with Solution**

**Problem **

Find the area of a rectangle whose length measures 4a^{2}b^{3} and width of 2a^{3}b^{4}.

**Solution**

Since the area of a rectangle is calculated using the formula AreaRectangle=Length×Width, when we substitute the given measurements, we get,

Area=4a^{2}b^{3} x 2a^{3}b^{4}

Applying the product rule of exponents, we have,

Area=(4⋅2)(a^{2+3})(b^{3+4})

Area=8a^{5}b^{7}

Hence, the area of the rectangle is 8a5b7.

**Practice Test**

Simplify each expression.

a. 2^{4}.2^{3}

b. $\frac{5^6}{5^2}$

c. (3^{2})^{3}

d. 7^{0}a^{4}

e. $\frac{3^5a^{-2}b^5}{b^4}$

*Answers:*

a. 2^{4}.2^{3}=2^{4+3}=2^{7}=128

b. $\frac{5^6}{5^2}$=5^{4}=625

c. (3^{2})^{3}=3^{2×3}=3^{6}=729

d. 7^{0}a^{4}=(1)a^{4} = a^{4}

e. $\frac{3^5a^{-2}b^5}{b^4}$ = $\frac{243b}{a^2}$

**Frequently Asked Questions (FAQs)**

**Why does any number to the power of zero equal one? **

According to the laws of exponents, when we divide any number by itself, we get 1.

For example, $\frac{2^3}{2^3}$ equals 1.

According to the quotient of powers rule, we can also express this as 2^{3-3}, which is 2^{0}. Therefore, any number to the power of zero is 1.

**Can zero be raised to an exponent? **

Zero can be raised to a positive exponent, and the result will always be zero. However, zero to the power of zero (0^{0}) is an indeterminate form, which means it does not have a clear, universally accepted value.

**What does a negative exponent mean? **

A negative exponent means that the base is on the wrong side of the fraction line, so you flip the base to the other side. For example, 2^{-3} equals $\frac{1}{2^3}$ or $\frac{1}{8}$.

**Why do we add exponents when multiplying like bases? **

When you multiply like bases, you add more base multiplication. The exponent indicates how many times a number is multiplied by itself, so when you multiply like bases, you add to the number of times the base is being multiplied, which is why the exponents are added.

**What happens if I raise a power to another power? **

When you raise a power to another power, you multiply the exponents. This is because you increase the number of times you multiply the base by itself.

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