**Multiplying Exponents With the Same Bases**

When multiplying exponents with like bases you simply add the exponents. The base does not change.

\(a^{b} \times a^{c} = a^{b + c}\)

Here's an example to show why this works.

\(3^{3} \times 3^{2}\)

\(=3 \times 3 \times\ 3 \times 3 \times 3 \)

\(=3^5\)

This is categorized under the product rules for exponents.

**Dividing Exponents With the Same Bases**

When dividing exponents with the same base you subtract the exponents. The base does not change.

\(a^{b} \div a^{c} = a^{b - c}\)

Here's an example to show why this works.

\(3^{4} \div 3^{2}\)

\(=3 \times 3 \times\ 3 \times 3 \div 3 \div 3\)

\(=3 \times 3\)

\(=3^2\)

This is categorized under the quotient rules for exponents

**Exponents to Exponents**

When you have a given base \(a\) raised to an exponent \(b\) in parentheses which is in turn raised to another exponent \(c\), or \((a^{b})^{c}\) you can multiply the two exponents \( b\) and \(c\) and the base stays the same. In other words \((a^{b})^{c}\)= \(a^{b \times c}\)

Although you can use this to simplify an exponent to an exponent such as

\((9^{2})^{3}\)= \(9^{2 \times 3}\) = \(9^{6}\)

Keep in mind that this works in reverse as well allowing you to factor out numbers from the exponents if necessary.

Be careful though because if there are no parentheses you must use a different rule.

When you have a given base \(a\) raised to an exponent \(b\) which is in turn raised to another exponent \(c\), or \( a^{b^{c}} \) you solve \( b\) to the exponent \(c\) first and the base stays the same. In other words \( a^{b^{c}} \)= \( a^ {(b^{ c})} \)

for example \( 9^{2^{3}} \)= \( 9^ {(2^ {3})} \) = \(9^{8}\)

These are called the power rules of exponents.

**Different Bases Same Exponents**

The Product Rule

\(a^{n} \times b^{n} = (a \times b)^{n}\)

The Quotient Rule

\(a^{n} \div b^{n} = (a \div b)^{n}\)

**Factoring out bases**

This is basically just the previous section reworded because this is a very important concept. If you have a base that is not a prime number you can divide factors out of it. Lets say we have a base \(d\) such that \(d=a \times b\) , then it is also true that \( d^{c} = a^{c} \times b^{c} \)

example 1

\(27^{3}\)

\(= (9 \times 3)^{3}\)

\(=9^{3} \times 3^{3}\)

example 2

\( =\sqrt{45} \)

\( =\sqrt{9 \times 5} \)

\( =\sqrt{9} \times \sqrt{5} \)

\( =3 \times \sqrt{5} \)

**Adding and Subtracting Exponents**

I do not know if there is a general rule for adding exponents I find the best way is to treat each problem uniquely. That being said using the exponent rules up above you can start to see different methods which allow you to simplify these types of expressions.

example 1

\( 27^{2} + 9^{2} \)

\( = (9 \times 3)^{2} + 9^{2} \)

\( = 9^{2} \times 3^{2} + 9^{2} \)

\( = 9^{2} \times 9 + 9^{2} \)

\( = 9(9^{2})+ 1(9^{2}) \)

\( = 10 (9^{2}) \)

This can be further simplifed.

\( = 10 ((3^{2})^{2} ) \)

\( = 10 (3^{2 +2 }) \)

\( = 10(3^{4}) \)

example 2

\(5 \sqrt{44} + 2 \sqrt{11} \)

\( = 5 \sqrt{4 \times 11} + 2 \sqrt{11} \)

\( = 5\times \sqrt{4} \times \sqrt{11} + 2 \sqrt{11} \)

\( =(5 \times 2) \sqrt{11} + 2 \sqrt{11} \)

\( = 10 \sqrt{11} + 2 \sqrt {11} \)

\( = 12 \sqrt{11} \)

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