We are asked how properties properties of integer exponents help us write equivalent expressions. To answer this question, let's take a look at the properties of integer exponents that we know. The first one is the Product of Powers Property.

Product of Powers Property
When the bases are the same in multiplication, add the exponents. $a^{m} \cdot a^{n} = a^{m + n}$

As we can see, this property tells us that if we multiply two powers with the same base, then we can add the exponents. This means that the expression

8^{2} \cdot 8^{5}

is equivalent to

8^{7} .

8^{2} \cdot 8^{5} = 8^{2 + 5} = 8^{7}

The next property is the Power of Products Property.

Power of Products Property
When the bases are different and the exponents are the same in multiplication, multiply the bases and keep the exponent the same. $a^{n} \cdot b^{n} = (a \cdot b)^{n}$

This property tells us that if we multiply two exponential expressions with the same exponent and different bases, then we can multiply the bases and keep the exponent the same. This means that the expression

5^{4} \cdot 4^{4}

is equivalent to

2 0^{4} .

5^{4} \cdot 4^{4} = (5 \cdot 4)^{4} = 2 0^{4}

The next property is the Power of Powers Property.

Power of Powers Property
To find the power of a power, multiply the exponents. $(a^{m})^{n} = a^{m \cdot n}$

As we can see, it tells us that to find the power of a power, we multiply the exponents. This means that the expression

(6^{2})^{4}

is equivalent to

6^{8} .

(6^{2})^{4} = 6^{2 \cdot 4} = 6^{8}

The final property is the Quotient of Powers Property.

Quotient of Powers Property
When the bases are the same in division, subtract the exponents. $a^{m} \div a^{n} = a^{m - n}$

We can see that this property tells us that when dividing two exponential expressions with the same base, we subtract the exponents. This means that the expression

2^{7} \div 2^{4}

is equivalent to

2^{3} .

2^{7} \div 2^{4} = 2^{7 - 4} = 2^{3}

Exercise