## 17Calculus Precalculus - Exponent Rules

In calculus you will be required to understand and be able to use the rules of exponents. Most instructors will not go over these rules before jumping into calculus. So make sure you know them well.

List of Rules

Let's start by going through these exponent rules.

 1. The Product Rule $$x^n \cdot x^m = x^{n+m}$$ 2. The Quotient Rule $$\displaystyle{ \frac{x^n}{x^m} = x^{n-m} }$$ 3. The Power Rule $$(x^n)^m = x^{n \cdot m}$$ 4. Power of Zero $$x^0 = 1$$ 5. Negative Exponents $$\displaystyle{ x^{-n} = \frac{1}{x^n} }$$ 6. Fractional Exponents $$x^{1/n} = \sqrt[n]{x}$$ 7. Distribute An Exponent Over a Product $$(x \cdot y)^n = x^n y^n$$ 8. Distribute An Exponent Over a Quotient $$\displaystyle{ \left( \frac{x}{y} \right)^n = \frac{x^n}{y^n} }$$

In this video clip, the instructor explains these rules very well and includes some examples.

### freeCodeCamp.org - College Algebra - Exponent Rules [10min-13secs]

video by freeCodeCamp.org

This instructor also does a great job explaining these rules. Before starting the practice problems, watch this video also to make sure you are ready.

### Mario's Math Tutoring - Exponent Rules with Examples [6min-40secs]

Negative exponents can be one of the most difficult concepts when simplifying exponents. Here is a good video clip explaining negative exponents.

### The Organic Chemistry Tutor - Negative Exponents Explained! [43secs]

Okay, let's practice these rules on these problems.

Practice Exponent Rules

$$x^{-3}$$

Problem Statement

Simplify $$x^{-3}$$ using the exponent rules, giving your answer in exact form.

$$1/x^3$$

Problem Statement

Simplify $$x^{-3}$$ using the exponent rules, giving your answer in exact form.

Solution

### 3825 video

$$1/x^3$$

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$$\displaystyle{ \frac{1}{x^{-4}} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{1}{x^{-4}} }$$ using the exponent rules, giving your answer in exact form.

$$x^4$$

Problem Statement

Simplify $$\displaystyle{ \frac{1}{x^{-4}} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3826 video

$$x^4$$

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Simplify $$- 3^2$$ and $$(-3)^2$$

Problem Statement

Simplify $$- 3^2$$ and $$(-3)^2$$ using the exponent rules, giving your answer in exact form.

$$- 3^2 = -9$$ and $$(-3)^2 = 9$$

Problem Statement

Simplify $$- 3^2$$ and $$(-3)^2$$ using the exponent rules, giving your answer in exact form.

Solution

### 3827 video

$$- 3^2 = -9$$ and $$(-3)^2 = 9$$

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$$x^4 \cdot x^{-9}$$

Problem Statement

Simplify $$x^4 \cdot x^{-9}$$ using the exponent rules, giving your answer in exact form.

$$1/x^5$$

Problem Statement

Simplify $$x^4 \cdot x^{-9}$$ using the exponent rules, giving your answer in exact form.

Solution

### 3828 video

$$1/x^5$$

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$$\displaystyle{ \left( \frac{3}{4} \right)^{-2} }$$

Problem Statement

Simplify $$\displaystyle{ \left( \frac{3}{4} \right)^{-2} }$$ using the exponent rules, giving your answer in exact form.

$$16/9$$

Problem Statement

Simplify $$\displaystyle{ \left( \frac{3}{4} \right)^{-2} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3808 video

$$16/9$$

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$$(x^3)^4$$

Problem Statement

Simplify $$(x^3)^4$$ using the exponent rules, giving your answer in exact form.

$$x^{12}$$

Problem Statement

Simplify $$(x^3)^4$$ using the exponent rules, giving your answer in exact form.

Solution

### 3824 video

$$x^{12}$$

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$$(3x^2)^3$$

Problem Statement

Simplify $$(3x^2)^3$$ using the exponent rules, giving your answer in exact form.

$$27x^6$$

Problem Statement

Simplify $$(3x^2)^3$$ using the exponent rules, giving your answer in exact form.

Solution

### 3832 video

$$27x^6$$

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$$(x^3)^{-4}$$

Problem Statement

Simplify $$(x^3)^{-4}$$ using the exponent rules, giving your answer in exact form.

$$1/x^{12}$$

Problem Statement

Simplify $$(x^3)^{-4}$$ using the exponent rules, giving your answer in exact form.

Solution

### 3809 video

$$1/x^{12}$$

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$$x^4 \cdot x^5$$

Problem Statement

Simplify $$x^4 \cdot x^5$$ using the exponent rules, giving your answer in exact form.

$$x^9$$

Problem Statement

Simplify $$x^4 \cdot x^5$$ using the exponent rules, giving your answer in exact form.

Solution

### 3822 video

$$x^9$$

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$$(x^3)^{-5}$$

Problem Statement

Simplify $$(x^3)^{-5}$$ using the exponent rules, giving your answer in exact form.

$$1/x^{15}$$

Problem Statement

Simplify $$(x^3)^{-5}$$ using the exponent rules, giving your answer in exact form.

Solution

### 3829 video

$$1/x^{15}$$

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$$x^3 \cdot x^{-8}$$

Problem Statement

Simplify $$x^3 \cdot x^{-8}$$ using the exponent rules, giving your answer in exact form.

$$1/x^5$$

Problem Statement

Simplify $$x^3 \cdot x^{-8}$$ using the exponent rules, giving your answer in exact form.

Solution

### 3810 video

$$1/x^5$$

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$$(4x^{-5})^2$$

Problem Statement

Simplify $$(4x^{-5})^2$$ using the exponent rules, giving your answer in exact form.

$$16/x^{10}$$

Problem Statement

Simplify $$(4x^{-5})^2$$ using the exponent rules, giving your answer in exact form.

Solution

### 3811 video

$$16/x^{10}$$

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$$\displaystyle{ \frac{x^7}{x^3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^7}{x^3} }$$ using the exponent rules, giving your answer in exact form.

$$x^4$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^7}{x^3} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3823 video

$$x^4$$

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$$\displaystyle{ \frac{x^7}{x^{12}} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^7}{x^{12}} }$$ using the exponent rules, giving your answer in exact form.

$$1/x^5$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^7}{x^{12}} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3830 video

$$1/x^5$$

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$$\displaystyle{ \frac{y^5}{y^{-3}} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{y^5}{y^{-3}} }$$ using the exponent rules, giving your answer in exact form.

$$y^8$$

Problem Statement

Simplify $$\displaystyle{ \frac{y^5}{y^{-3}} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3812 video

$$y^8$$

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$$\displaystyle{ \frac{x^{-8}}{x^{-11}} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^{-8}}{x^{-11}} }$$ using the exponent rules, giving your answer in exact form.

$$x^3$$

Problem Statement

Simplify $$\displaystyle{ \frac{x^{-8}}{x^{-11}} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3813 video

$$x^3$$

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$$\displaystyle{ \frac{b^{-4}}{b^5} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{b^{-4}}{b^5} }$$ using the exponent rules, giving your answer in exact form.

$$1/b^9$$

Problem Statement

Simplify $$\displaystyle{ \frac{b^{-4}}{b^5} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3819 video

$$1/b^9$$

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$$\displaystyle{ \frac{a^3}{a^7} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{a^3}{a^7} }$$ using the exponent rules, giving your answer in exact form.

$$1/a^4$$

Problem Statement

Simplify $$\displaystyle{ \frac{a^3}{a^7} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3820 video

$$1/a^4$$

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Simplifying Using Exponent Rules

In this video clip, the instructor shows examples of how to use the exponent rules to simplify expressions.

### freeCodeCamp.org - College Algebra - Simplifying [21min-32secs]

video by freeCodeCamp.org

Okay, let's practice simplifying on these problems.

Practice Simplifying

$$5-2(x^{-3}x^4)^0$$

Problem Statement

Simplify $$5-2(x^{-3}x^4)^0$$ using the exponent rules, giving your answer in exact form.

$$3$$

Problem Statement

Simplify $$5-2(x^{-3}x^4)^0$$ using the exponent rules, giving your answer in exact form.

Solution

Comment: Although in general, his answer is correct, he is making an assumption that $$x \neq 0$$. He doesn't need to state this since it is implied by the term $$x^{-3}$$. If $$x = 0$$ is part of the domain, this term would be $$0^{-3} = 1/0^3 = 1/0$$. Remember in algebra, we never want the denominator to be zero. So, in this case the domain does not include zero. So he doesn't have to worry about the case $$0^0$$ for this problem.

### 3814 video

$$3$$

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$$(-2x^3y^4)^2$$

Problem Statement

Simplify $$(-2x^3y^4)^2$$ using the exponent rules, giving your answer in exact form.

$$4x^6y^8$$

Problem Statement

Simplify $$(-2x^3y^4)^2$$ using the exponent rules, giving your answer in exact form.

Solution

### 3833 video

$$4x^6y^8$$

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$$(5x^3)(4x^7)$$

Problem Statement

Simplify $$(5x^3)(4x^7)$$ using the exponent rules, giving your answer in exact form.

$$20x^{10}$$

Problem Statement

Simplify $$(5x^3)(4x^7)$$ using the exponent rules, giving your answer in exact form.

Solution

### 3834 video

$$20x^{10}$$

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$$(7x^6)(5x^4)$$

Problem Statement

Simplify $$(7x^6)(5x^4)$$ using the exponent rules, giving your answer in exact form.

$$35x^{10}$$

Problem Statement

Simplify $$(7x^6)(5x^4)$$ using the exponent rules, giving your answer in exact form.

Solution

### 3835 video

$$35x^{10}$$

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$$(4x^3y^2)(7x^{-4}y^3)$$

Problem Statement

Simplify $$(4x^3y^2)(7x^{-4}y^3)$$ using the exponent rules, giving your answer in exact form.

$$\displaystyle{ \frac{28y^5}{x} }$$

Problem Statement

Simplify $$(4x^3y^2)(7x^{-4}y^3)$$ using the exponent rules, giving your answer in exact form.

Solution

### 3836 video

$$\displaystyle{ \frac{28y^5}{x} }$$

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$$(3x^{-2}y^3)^2 (4x^5y^{-6})^3$$

Problem Statement

Simplify $$(3x^{-2}y^3)^2 (4x^5y^{-6})^3$$ using the exponent rules, giving your answer in exact form.

$$\displaystyle{ \frac{576x^{11}}{y^{12}} }$$

Problem Statement

Simplify $$(3x^{-2}y^3)^2 (4x^5y^{-6})^3$$ using the exponent rules, giving your answer in exact form.

Solution

### 3816 video

$$\displaystyle{ \frac{576x^{11}}{y^{12}} }$$

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$$\displaystyle{ \frac{3a^{-4}}{b^{-6}c^3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{3a^{-4}}{b^{-6}c^3} }$$ using the exponent rules, giving your answer in exact form.

$$\displaystyle{ \frac{3b^6}{a^4c^3} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{3a^{-4}}{b^{-6}c^3} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3815 video

$$\displaystyle{ \frac{3b^6}{a^4c^3} }$$

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$$\displaystyle{ \frac{24x^7y^3}{8x^4y^{-12}} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{24x^7y^3}{8x^4y^{-12}} }$$ using the exponent rules, giving your answer in exact form.

$$3x^3y^{15}$$

Problem Statement

Simplify $$\displaystyle{ \frac{24x^7y^3}{8x^4y^{-12}} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3837 video

$$3x^3y^{15}$$

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$$\displaystyle{ \frac{45x^4y^{-6}}{27x^{-7}y^5} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{45x^4y^{-6}}{27x^{-7}y^5} }$$ using the exponent rules, giving your answer in exact form.

$$\displaystyle{ \frac{5x^{11}}{3y^{11}} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{45x^4y^{-6}}{27x^{-7}y^5} }$$ using the exponent rules, giving your answer in exact form.

Solution

1. Notice in his solution, he started with the constants and simplified them first. This is a great way to make the problem seem not so complicated and we highly recommend this technique.
2. Although his answer is correct, you could also write the final answer as $$(5/3) (x/y)^11$$

### 3817 video

$$\displaystyle{ \frac{5x^{11}}{3y^{11}} }$$

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$$\displaystyle{ \frac{18a^3b^{-4}c^{-6}}{30a^7b^5c^{-9}} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{18a^3b^{-4}c^{-6}}{30a^7b^5c^{-9}} }$$ using the exponent rules, giving your answer in exact form.

$$\displaystyle{ \frac{3c^3}{5a^4b^9} }$$

Problem Statement

Simplify $$\displaystyle{ \frac{18a^3b^{-4}c^{-6}}{30a^7b^5c^{-9}} }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3818 video

$$\displaystyle{ \frac{3c^3}{5a^4b^9} }$$

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$$\displaystyle{ \left( \frac{35x^3y^5}{63x^{-4}y^7} \right)^2 }$$

Problem Statement

Simplify $$\displaystyle{ \left( \frac{35x^3y^5}{63x^{-4}y^7} \right)^2 }$$ using the exponent rules, giving your answer in exact form.

$$\displaystyle{ \frac{25x^{14}}{81y^4} }$$

Problem Statement

Simplify $$\displaystyle{ \left( \frac{35x^3y^5}{63x^{-4}y^7} \right)^2 }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3831 video

$$\displaystyle{ \frac{25x^{14}}{81y^4} }$$

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$$\displaystyle{ \left( \frac{ 63x^{-3}y^4z^{-5} }{ 35x^8y^{-7}z^{-4} } \right)^2 }$$

Problem Statement

Simplify $$\displaystyle{ \left( \frac{ 63x^{-3}y^4z^{-5} }{ 35x^8y^{-7}z^{-4} } \right)^2 }$$ using the exponent rules, giving your answer in exact form.

$$\displaystyle{ \frac{81y^{22}}{25x^{22}z^2} }$$

Problem Statement

Simplify $$\displaystyle{ \left( \frac{ 63x^{-3}y^4z^{-5} }{ 35x^8y^{-7}z^{-4} } \right)^2 }$$ using the exponent rules, giving your answer in exact form.

Solution

### 3821 video

$$\displaystyle{ \frac{81y^{22}}{25x^{22}z^2} }$$

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