# Properties of Exponents Lesson

Properties of Exponents are extremely important in mathematics. We use the properties of exponents to simplify expressions with exponents. There are eight properties of exponents that will be discussed in this lesson. These properties also apply to expressions when the exponents include decimals, fractions, negative numbers, and irrational numbers. Below is the list of the properties of exponents:

**Zero Exponent Rule**– any non-zero number raised to the zero power is equal to 1.

a^{0} = 1

**Product Rule**– When you multiply two or more exponential expressions with the same base, you can add their exponents.

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

**Quotient Rule**– When you divide two exponential expressions with the same base, you can subtract their exponents. Subtract the exponent in the denominator from the exponent in the numerator.

a^{m}/a^{n} = a^{m – n}

**Power to a Power Rule**– When you raise an exponential expression to another exponent, you can multiply the exponents.

(a^{m})^{n} = a^{m x n}

**Negative Exponent Rule**– When an expression is raised to a negative exponent, you can convert the power to a positive exponent by taking the reciprocal of the expression.

a^{-m} = 1/a^{m}

6. **Product of Powers Rule**– When multiplying two terms with the same exponent, you can divide the bases and keep the exponents unchanged.

a^{m} x b^{m} = (ab)^{m}

**Quotient of Powers Rule**– When dividing two terms with the same exponent, you can divide the bases and keep the exponents the same.

a^{m}/b^{m} = (a/b)^{m}

**Fractional Exponents Rule**– A fractional exponent represents a root. When given a fractional exponent, you can take the denominator and write it as the index of the root, and the numerator is the exponent of the base. The expression with the numerator is written as the radicand.

Additionally we know that an expression raised to the power of one is the given expression.

a^{1} = a

Let’s look at specific examples for each of the properties above to gain a better understanding.

**Zero Exponent Rule**

Simplify the expression 4^{0}.

Due to the zero exponent rule we know that any non-zero number raised to the power of zero is equal to one. Therefore, 4^{0} = 1. This can be better understood following the example below.

**Product Rule**

Simplify the expression 4^{3} x 4^{2}

Using the product rule we know that since the bases are the same, both 4, we can add the exponents.

Therefore, 4^{3} x 4^{2} = 4^{3+2} = 4^{5}. Now let’s look at the same example to understand why the product rule works.

4^{3} x 4^{2} = (4 x 4 x 4) x (4 x 4) = 4^{5}

**Quotient Rule**

Simplify the expression: 4^{5} x 4^{2}.

Using the quotient rule we know that since the bases are the same, both 4, we can subtract the exponents.

Therefore, 4^{5} / 4^{2} = 4^{5 – 2} = 4^{3} . This can be better understood following the example below.

**Power to a Power Rule**

Simplify the expression (4^{2})^{3}.

Using the power to a power rule we know that since the bases are the same, both 4, we can multiply the exponents together. Therefore, (4^{2})^{3} = 4^{2 x 3} = 4^{6}. This can be better understood following the example below.

(4^{2})^{3} = (4 x 4)^{3} = (4 x 4) (4 x 4) (4 x 4) = 4^{6}

**Negative Exponent Rule**

Simplify the expression 4^{-3}.

Using the negative exponent rule we know that a negative exponent means we need to take the reciprocal of the expression and make the exponent positive. Therefore, 4^{-3} = 1/4^{3}.

**Product of Powers Rule**

Simplify the expression 4^{2} x 3^{2}

Using the product of powers rule we know that since the exponents are the same, both 2, we can multiply the bases and raise the product to the second power. Therefore, 4^{2} x 3^{2} = (4 x 3)^{2} = 12^{2} = 144. This can be better understood following the example below.

4^{2} x 3^{2} = 4 x 4 x 3 x 3 = (4 x 3) x (4 x 3) = 12 x 12 = 144.

**Quotient of Powers Rule**

Simplify the expression: 4^{2} / 2^{2}

Using the quotient of powers rule we know that since the exponents are the same, both 2, we can divide the numerator and denominator and raise the quotient to the second power. Therefore, 4^{2} / 2^{2} = (4/2)^{2} = 2^{2} = 4

This can be better understood following the example below.

**Fractional Exponent Rule**

Simplify the expression 4^{3/2}

Using the fractional exponent rule we know that the numerator, 3, is the exponent of the base and the denominator, 2, is the index of the root.

from following the example below.

**Examples**

**Example 1: **Simplify the expression a^{2} x a^{4} x a^{5} using the properties of exponents.

**Solution:** First notice that the base of each of the exponents is the same, then notice that we are multiplying each term together. This means we will use the Product Property which states we can add the exponents.

Therefore,a^{2} x a^{4} x a^{5} = a^{2 + 4 + 5} = a^{11}

The answer is a^{11}

**Example 2: **Use the properties of exponents to simplify x^{7} / x^{4}

**Solution:** Start by noticing that the bases of both the numerator and denominator are both x, and this is division, therefore we will use the quotient rule of exponents. This means that we will subtract our exponents, as shown below.

The answer is x^{3}

**Example 3: **Simplify the expression using properties of exponents (m^{4})^{2} x m^{3}

**Solution: **First we will need to apply the power to a power rule to eliminate the parentheses, to do this we multiply the exponents 4 and 2, which equals 8. Then we will apply the product rule since we will have m^{8} x m^{3}. The product rule states that we can add the exponents if we are multiplying two bases that are the same.

Therefore,m^{8} x m^{3} = m^{8 + 3} = m^{11}.

The answer is m^{11}.

FAQs on Properties of Exponents

**1) What are exponents?**

Exponents represent repeated multiplication of the same number. They are written as a superscript to the right of the base number. For example, in 43, 4 is the base and 3 is the exponent.

**2) How do I simplify expressions with exponents?**

To simplify expressions with exponents, apply the properties of exponents listed above. Look for opportunities to combine like terms, eliminate negative exponents, and simplify fractions.

**3) What does a zero exponent mean?**

A zero exponent means that the base raised to the power of zero equals 1. So, a0 = 1, where a is any non-zero real number.

**4) Can exponents be applied to non-integer numbers?**

Yes,exponents can be applied to non-integer numbers. For example, you can have expressions like 3.64or (0.6)-4. The properties of exponents still remain true for non-integer numbers.

**5) How do I simplify expressions with variables as bases?**

When simplifying expressions with variables as bases, apply the same properties of exponents. Keep in mind that variables with the same base can be combined by adding or subtracting their exponents.

**6) Are there any special cases when working with exponents?**

One special case is when the base is 1. Any non-zero number raised to the power of 0 is 1, but 1 raised to any power remains 1. Additionally, any number raised to the power of 1 is itself.

**7) Why are properties of exponents important?**

Understanding the properties of exponents allows for the manipulation and simplification of expressions involving exponents, which is crucial in various mathematical contexts including algebra, calculus, and physics. These properties provide shortcuts for simplifying complex expressions and solving equations efficiently.