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Lesson Explainer: Simplifying Expressions: Rules of Exponents Mathematics • 9th Grade

In this explainer, we will learn how to simplify algebraic expressions using the rules of exponents.

An algebraic expression will behave in a very similar manner to an expression involving numbers. This is because the unknown variables in an algebraic expression are unknown numbers. We can use this idea to simplify algebraic expressions involving exponents; we can consider how we would simplify an expression involving exponents of numbers.

For example, consider 22๏Šซ๏Šฉ. We can evaluate this using a calculator; however, we can also do this by hand. We recall that raising a number ๐‘ to a positive integer exponent ๐‘› means multiplying ๐‘› lots of ๐‘. We can use this to expand the powers as products. We know that 2๏Šซ is the product of 5 lots of 2 and 2๏Šฉ is the product of 3 lots of 2: 2=2ร—2ร—2ร—2ร—2,2=2ร—2ร—2.๏Šซ๏Šฉ

Substituting these into quotient gives us 22=2ร—2ร—2ร—2ร—22ร—2ร—2.๏Šซ๏Šฉ

We can now cancel three of the shared factors of 2 in the numerator and denominator to get

We can apply this same process even if we do not know the base of the expressions. For example, if we have ๐‘๐‘๏Šซ๏Šฉ, then we can follow the exact same reasoning: ๐‘๐‘=๐‘ร—๐‘ร—๐‘ร—๐‘ร—๐‘๐‘ร—๐‘ร—๐‘=๐‘=๐‘.๏Šซ๏Šฉ๏Šซ๏Šฑ๏Šฉ๏Šจ

We can follow this same reasoning to show many useful results in simplifying exponential expressions. These are known as the laws of exponents.

Law: Laws of Exponents

For any bases ๐‘Ž and ๐‘ and positive integers ๐‘š and ๐‘›, we have the following:

  • The product rule for exponents: ๐‘ร—๐‘=๐‘๏‰๏Š๏‰๏Šฐ๏Š
  • The quotient rule for exponents: ๐‘๐‘=๐‘๏‰๏Š๏‰๏Šฑ๏Š
  • The power of a power rule for exponents: (๐‘)=๐‘๏‰๏Š๏‰ร—๏Š
  • The power of a product rule for exponents: (๐‘Ž๐‘)=๐‘Ž๐‘๏Š๏Š๏Š

It is worth noting that these rules always hold true provided that we can evaluate the expressions. For example, we cannot divide by 0, so we cannot apply the quotient rule for exponents if the base is 0.

We can actually prove all of these laws by expanding the products and using the properties of powers and products. For example, we can rewrite ๐‘ร—๐‘๏‰๏Š by expanding each power:

We have a product of ๐‘š factors of ๐‘ and ๐‘› factors of ๐‘. We can note that this is the product of ๐‘š+๐‘› factors of ๐‘, which can be rewritten as ๐‘๏‰๏Šฐ๏Š:

We can follow this same process to show all of these laws.

In our first example, we will use the laws of exponents to simplify the product of two monomials.

Example 1: Simplifying the Product of Two Algebraic Expressions

Simplify ๐‘ฅร—๐‘ฅ๏Šจ๏Šฉ.

Answer

To simplify the product of two monomials, we first recall that the product rule for exponents tells us that ๐‘ร—๐‘=๐‘๏‰๏Š๏‰๏Šฐ๏Š. We can apply this rule with ๐‘=๐‘ฅ, ๐‘š=2, and ๐‘›=3 to get ๐‘ฅร—๐‘ฅ=๐‘ฅ=๐‘ฅ.๏Šจ๏Šฉ๏Šจ๏Šฐ๏Šฉ๏Šซ

In our next example, we will use the laws of exponents to simplify the quotient of two monomials.

Example 2: Simplifying the Division of Two Algebraic Expressions

Simplify ๐‘ฅ๐‘ฅ๏Šญ๏Šฌ.

Answer

We are asked to simplify the quotient of two monomials. We can do this by recalling that this is similar to the quotient rule for exponents, which says that for positive integers ๐‘š and ๐‘›๐‘๐‘=๐‘.๏‰๏Š๏‰๏Šฑ๏Š

In this case, we have ๐‘=๐‘ฅ, ๐‘š=7, and ๐‘›=6. Thus, the new exponent is found by subtracting the powers: ๐‘ฅ๐‘ฅ=๐‘ฅ=๐‘ฅ.๏Šญ๏Šฌ๏Šญ๏Šฑ๏Šฌ๏Šง

Finally, we recall that raising a number to the power of 1 leaves it unchanged. So, ๐‘ฅ=๐‘ฅ๏Šง, and we have shown that ๐‘ฅ๐‘ฅ=๐‘ฅ.๏Šญ๏Šฌ

In our next example, we will simplify the product of two monomials by using the laws of exponents.

Example 3: Simplifying the Multiplication of Two Algebraic Expressions

Simplify 7๐‘ฅร—8๐‘ฅ๏Šช๏Šญ.

Answer

We are asked to simplify the product of two monomials. We can do this by noting that this is similar to the product rule for exponents, which says that for positive integers ๐‘š and ๐‘›ย  ๐‘ร—๐‘=๐‘.๏‰๏Š๏‰๏Šฐ๏Š

We cannot directly apply this law since there are also constant factors. Instead, letโ€™s start by rearranging the product so that the like terms are multiplied together: 7๐‘ฅร—8๐‘ฅ=(7ร—8)ร—๏€น๐‘ฅร—๐‘ฅ๏….๏Šช๏Šญ๏Šช๏Šญ

We can calculate that 7ร—8=56 and we can simplify ๐‘ฅร—๐‘ฅ๏Šช๏Šญ by using the product rule for exponents. We have ๐‘=๐‘ฅ, ๐‘š=4, and ๐‘›=7: (7ร—8)ร—๏€น๐‘ฅร—๐‘ฅ๏…=56๏€น๐‘ฅ๏…=56๐‘ฅ.๏Šช๏Šญ๏Šช๏Šฐ๏Šญ๏Šง๏Šง

In our next example, we will use the laws of exponents to simplify the quotient of a polynomial and a monomial.

Example 4: Dividing a Polynomial by a Monomial

Simplify โˆ’43๐‘ฅ+12๐‘ฅโˆ’6๐‘ฅ๐‘ฅ๏Šญ๏Šฉ๏Šจ.

Answer

In order to simplify the quotient of a polynomial and a monomial, we first need to split the division over each term in the numerator: โˆ’43๐‘ฅ+12๐‘ฅโˆ’6๐‘ฅ๐‘ฅ=โˆ’43๐‘ฅ๐‘ฅ+12๐‘ฅ๐‘ฅโˆ’6๐‘ฅ๐‘ฅ.๏Šญ๏Šฉ๏Šจ๏Šญ๏Šฉ๏Šจ

We now see that each term is the quotient of monomials. We can simplify each term by using the quotient rule for exponents, which says that ๐‘๐‘=๐‘๏‰๏Š๏‰๏Šฑ๏Š. We recall that ๐‘ฅ=๐‘ฅ๏Šง, and we rewrite each term as follows: โˆ’43๐‘ฅ๐‘ฅ+12๐‘ฅ๐‘ฅโˆ’6๐‘ฅ๐‘ฅ=โˆ’43๏€พ๐‘ฅ๐‘ฅ๏Š+12๏€พ๐‘ฅ๐‘ฅ๏Šโˆ’6๏€พ๐‘ฅ๐‘ฅ๏Š.๏Šญ๏Šฉ๏Šจ๏Šญ๏Šง๏Šฉ๏Šง๏Šจ๏Šง

We can now simplify each quotient by using the quotient rule for exponents. We have โˆ’43๏€พ๐‘ฅ๐‘ฅ๏Š+12๏€พ๐‘ฅ๐‘ฅ๏Šโˆ’6๏€พ๐‘ฅ๐‘ฅ๏Š=โˆ’43๐‘ฅ+12๐‘ฅโˆ’6๐‘ฅ=โˆ’43๐‘ฅ+12๐‘ฅโˆ’6๐‘ฅ=โˆ’43๐‘ฅ+12๐‘ฅโˆ’6๐‘ฅ.๏Šญ๏Šง๏Šฉ๏Šง๏Šจ๏Šง๏Šญ๏Šฑ๏Šง๏Šฉ๏Šฑ๏Šง๏Šจ๏Šฑ๏Šง๏Šฌ๏Šจ๏Šง๏Šฌ๏Šจ

In our next example, we will simplify an algebraic expression that involves distributing the product of a monomial over a binomial.

Example 5: Multiplying a Monomial by a Binomial and Simplifying

Calculate โˆ’2๐‘ฅ๏€น๐‘ฅ+๐‘ฅ๏…๏Šจ๏Šช๏Šจ.

Answer

We want to expand the product of a monomial and a binomial. We can do this by distributing the product over each term in the binomial. We do this in exactly the same way that we would distribute the product of a constant over a binomial:

We can simplify each term by noting that they each include a product of powers. We can use the product rule for exponents, which states that ๐‘ร—๐‘=๐‘๏‰๏Š๏‰๏Šฐ๏Š.

We rewrite each term to show the product of the powers and apply the laws of exponents: ๏€น๏€นโˆ’2๐‘ฅ๏…ร—๐‘ฅ๏…+๏€น๏€นโˆ’2๐‘ฅ๏…ร—๐‘ฅ๏…=โˆ’2๏€น๐‘ฅร—๐‘ฅ๏…โˆ’2๏€น๐‘ฅร—๐‘ฅ๏…=โˆ’2๐‘ฅโˆ’2๐‘ฅ=โˆ’2๐‘ฅโˆ’2๐‘ฅ.๏Šจ๏Šช๏Šจ๏Šจ๏Šจ๏Šช๏Šจ๏Šจ๏Šจ๏Šฐ๏Šช๏Šจ๏Šฐ๏Šจ๏Šฌ๏Šช

In our final example, we will apply multiple laws of exponents to simplify an algebraic expression.

Example 6: Simplifying an Expression Using Rules of Exponents

Simplify ๏€น2๐‘ฅ๏…4๐‘ฅร—๐‘ฅ๏Šจ๏Šช๏Šจ๏Šฉ.

Answer

We can simplify this expression by first simplifying the numerator and denominator separately using the laws of exponents. Letโ€™s start with the numerator. We note that this is a power of a product, so we will use the rule (๐‘Ž๐‘)=๐‘Ž๐‘๏Š๏Š๏Š with ๐‘Ž=2, ๐‘=๐‘ฅ๏Šจ, and ๐‘›=4: ๏€น2๐‘ฅ๏…=2๏€น๐‘ฅ๏….๏Šจ๏Šช๏Šช๏Šจ๏Šช

We can simplify this further by noting that ๏€น๐‘ฅ๏…๏Šจ๏Šช is the power of a power. We can use the law of exponents, which states that (๐‘)=๐‘๏‰๏Š๏‰ร—๏Š to simplify this: 2๏€น๐‘ฅ๏…=16๐‘ฅ=16๐‘ฅ.๏Šช๏Šจ๏Šช๏Šจร—๏Šช๏Šฎ

In the denominator, we have a product of powers, so we can simplify this using the law of exponents, which states that ๐‘ร—๐‘=๐‘๏‰๏Š๏‰๏Šฐ๏Š. We have 4๐‘ฅร—๐‘ฅ=4๏€น๐‘ฅร—๐‘ฅ๏…=4๐‘ฅ=4๐‘ฅ.๏Šจ๏Šฉ๏Šจ๏Šฉ๏Šจ๏Šฐ๏Šฉ๏Šซ

We can now substitute the expressions for the numerator and denominator into the quotient to get ๏€น2๐‘ฅ๏…4๐‘ฅร—๐‘ฅ=16๐‘ฅ4๐‘ฅ.๏Šจ๏Šช๏Šจ๏Šฉ๏Šฎ๏Šซ

We can cancel the shared factor of 4 and use the quotient rule for exponents ๐‘๐‘=๐‘๏‰๏Š๏‰๏Šฑ๏Š to simplify this further. We rewrite the quotient to make the application of these rules easier: 16๐‘ฅ4๐‘ฅ=164ร—๐‘ฅ๐‘ฅ.๏Šฎ๏Šซ๏Šฎ๏Šซ

Now, we evaluate each quotient. We have 164ร—๐‘ฅ๐‘ฅ=4ร—๐‘ฅ=4๐‘ฅ.๏Šฎ๏Šซ๏Šฎ๏Šฑ๏Šซ๏Šฉ

Letโ€™s finish by recapping some of the important points from this explainer.

Key Points

  • For any bases ๐‘Ž and ๐‘ and positive integers ๐‘š and ๐‘›, we have the following:
    • The product rule for exponents: ๐‘ร—๐‘=๐‘๏‰๏Š๏‰๏Šฐ๏Š
    • The quotient rule for exponents: ๐‘๐‘=๐‘๏‰๏Š๏‰๏Šฑ๏Š
    • The power of a power rule for exponents: (๐‘)=๐‘๏‰๏Š๏‰ร—๏Š
    • The power of a product rule for exponents: (๐‘Ž๐‘)=๐‘Ž๐‘๏Š๏Š๏Š
    These all hold true provided that the expressions are well-defined.
  • We can expand the product of algebraic expressions and then apply the laws of exponents to find an equivalent expression for the product. Similarly, we can use the laws of exponents to find equivalent expressions for quotients of algebraic expressions.

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