Lesson Explainer: Powers of Monomials Mathematics

In this explainer, we will learn how to simplify algebraic expressions as monomials involving single and multiple variables using the power of a power property.

Let us start by recalling the power rule for exponents.

Power Rule for Exponents

If you raise a power to a power, this is equivalent to raising to the product of the two powers; that is, (๐‘Ž)=๐‘Ž.๏Š๏‰๏Šร—๏‰

We can use this rule to simplify expressions such as ๏€น๐‘ฆ๏…,๏Šฉ๏Šง๏Šฆ which could be written as ๐‘ฆ=๐‘ฆ.๏Šฉร—๏Šง๏Šฆ๏Šฉ๏Šฆ

Here, we are going to look at how we simplify a rational expression raised to a power. For example, ๏€พ๐‘ฅ๐‘ฆ๐‘ง๏Š.๏Šฉ๏Šช๏Šซ๏Šช

We can expand the expression to get ๐‘ฅ๐‘ฆ๐‘งร—๐‘ฅ๐‘ฆ๐‘งร—๐‘ฅ๐‘ฆ๐‘งร—๐‘ฅ๐‘ฆ๐‘ง,๏Šฉ๏Šช๏Šซ๏Šฉ๏Šช๏Šซ๏Šฉ๏Šช๏Šซ๏Šฉ๏Šช๏Šซ which is then the same as ๏€น๐‘ฅ๐‘ฆ๏…(๐‘ง).๏Šฉ๏Šช๏Šช๏Šซ๏Šช

We can now apply the product rule of exponents, (๐‘Ž๐‘)=๐‘Ž๐‘,๏Š๏Š๏Š to rewrite the numerator as follows: ๏€น๐‘ฅ๏…๏€น๐‘ฆ๏…(๐‘ง).๏Šฉ๏Šช๏Šช๏Šช๏Šซ๏Šช

At this point, we can apply the power rule for exponents to the top and bottom of the expression to get ๐‘ฅ๐‘ฆ๐‘ง,๏Šฉร—๏Šช๏Šชร—๏Šช๏Šซร—๏Šช which simplifies to ๐‘ฅ๐‘ฆ๐‘ง.๏Šง๏Šจ๏Šง๏Šฌ๏Šจ๏Šฆ

A key step to note in this process is how raising a quotient to a power ๐‘› is the same as raising the numerator to the power ๐‘› and raising the denominator to the power ๐‘›. We call this result the power of a quotient property.

Power of a Quotient

For any rational expression ๐‘Ž๐‘, where ๐‘โ‰ 0, ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘.๏Š๏Š๏Š

Let us now look at a few examples.

Example 1: Simplifying Powers of Rational Expressions

Simplify ๏€พ๐‘ฅ๐‘ฆ๏Š๏Šฉ๏Šช๏Šช.

Answer

With this question, we start by using the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š to rewrite the expression as ๏€น๐‘ฅ๏…(๐‘ฆ).๏Šฉ๏Šช๏Šช๏Šช

Using the power rule of exponents, (๐‘ฅ)=๐‘ฅ,๏Œบ๏Œป๏Œบร—๏Œป we can simplify the expression: the top simplifies to ๐‘ฅ=๐‘ฅ๏Šฉร—๏Šช๏Šง๏Šจ and the bottom to ๐‘ฆ=๐‘ฆ.๏Šชร—๏Šช๏Šง๏Šฌ Our final answer is, therefore, ๐‘ฅ๐‘ฆ.๏Šง๏Šจ๏Šง๏Šฌ

Example 2: Simplifying Powers of Rational Expressions Containing Products of Variables

Simplify ๏€พ๐‘ฅ๐‘ฆ๐‘ง๏Š๏Šฉ๏Šจ๏Šฉ๏Šฉ.

Answer

Our first step in answering this question is to use the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š to rewrite the expression as follows: ๏€น๐‘ฅ๐‘ฆ๏…(๐‘ง).๏Šฉ๏Šจ๏Šฉ๏Šฉ๏Šฉ

If we then recall the power rule of exponents, which tells us that (๐‘ฅ)=๐‘ฅ,๏Œบ๏Œป๏Œบร—๏Œป we can rewrite our expression as ๐‘ฅ๐‘ฆ๐‘ง.๏Šฉร—๏Šฉ๏Šจร—๏Šฉ๏Šฉร—๏Šฉ

We can now simplify all of the exponents to get ๐‘ฅ๐‘ฆ๐‘ง.๏Šฏ๏Šฌ๏Šฏ

Example 3: Simplifying Powers of Rational Expressions Containing Products of Variables

Simplify ๏€พ๐‘Ž๐‘๐‘๐‘‘๏Š๏Šจ๏Šช๏Šซ๏Šฌ๏Šฉ.

Answer

Our first step in answering this question is to use the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š to rewrite the expression as follows: ๏€น๐‘Ž๐‘๏…(๐‘๐‘‘).๏Šจ๏Šช๏Šฉ๏Šซ๏Šฌ๏Šฉ

If we then recall the power rule of exponents, which tells us that (๐‘ฅ)=๐‘ฅ,๏Œบ๏Œป๏Œบร—๏Œป we can rewrite our expression as ๐‘Ž๐‘๐‘๐‘‘.๏Šจร—๏Šฉ๏Šชร—๏Šฉ๏Šซร—๏Šฉ๏Šฌร—๏Šฉ

We can now simplify all of the exponents to get ๐‘Ž๐‘๐‘๐‘‘.๏Šฌ๏Šง๏Šจ๏Šง๏Šซ๏Šง๏Šฎ

Example 4: Simplifying Powers of Rational Expressions Containing Products of Variables

Simplify ๏€น๐‘Žรท๐‘๏…๏Šญ๏Šง๏Šฆ.

Answer

We begin by rewriting the expression as a quotient: ๏€น๐‘Žรท๐‘๏…=๏€ป๐‘Ž๐‘๏‡.๏Šญ๏Šง๏Šฆ๏Šญ๏Šง๏Šฆ We can now use the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š to rewrite the expression as follows: ๏€น๐‘Žรท๐‘๏…=๐‘Ž(๐‘).๏Šญ๏Šง๏Šฆ๏Šง๏Šฆ๏Šญ๏Šง๏Šฆ

Using the power rule for exponents, we can further simplify the denominator of the expression as follows: ๏€น๐‘Žรท๐‘๏…=๐‘Ž๐‘=๐‘Ž๐‘.๏Šญ๏Šง๏Šฆ๏Šง๏Šฆ๏Šญร—๏Šง๏Šฆ๏Šง๏Šฆ๏Šญ๏Šฆ

Example 5: Simplifying Powers of Rational Expressions Containing Products of Variables

Write an equivalent expression to ๏€พ๐‘ฅ๐‘ฆ๐‘ง๏Š๏Šญ๏Šซ๏Šซ๏Šฉ that does not include parentheses.

Answer

Our first step in answering this question is to use the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š to rewrite the expression as follows: ๏€น๐‘ฅ๐‘ฆ๏…(๐‘ง).๏Šญ๏Šซ๏Šฉ๏Šซ๏Šฉ

We can now use the product rule for exponents to rewrite the numerator as follows: ๏€น๐‘ฅ๏…๏€น๐‘ฆ๏…(๐‘ง).๏Šญ๏Šฉ๏Šซ๏Šฉ๏Šซ๏Šฉ

At this point, we can use the power rule of exponents, which tells us that (๐‘ฅ)=๐‘ฅ,๏Œบ๏Œป๏Œบร—๏Œป to rewrite our expression as follows: ๐‘ฅ๐‘ฆ๐‘ง.๏Šญร—๏Šฉ๏Šซร—๏Šฉ๏Šซร—๏Šฉ

Finally, we can now simplify all of the exponents to get ๐‘ฅ๐‘ฆ๐‘ง.๏Šจ๏Šง๏Šง๏Šซ๏Šง๏Šซ

We can also simplify products of powers of rational expressions using very similar methods. Generally, this type of question will include an additional step of simplification once we have dealt with any exponents. Let us look at an example of this now.

Example 6: Simplifying Products of Powers of Rational Expressions

Simplify ๏€พ๐‘Ž๐‘๏Šร—๏€พ๐‘๐‘Ž๏Š๏Šจ๏Šซ๏Šช๏Šง๏Šฆ๏Šช๏Šจ.

Answer

With this question, we can start by using the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏Š๏Š๏Š to rewrite the expression as follows: ๏€น๐‘Ž๏…(๐‘)ร—๏€น๐‘๏…(๐‘Ž).๏Šจ๏Šช๏Šซ๏Šช๏Šง๏Šฆ๏Šจ๏Šช๏Šจ If we then recall the power rule of exponents, which tells us that (๐‘ฅ)=๐‘ฅ,๏Œบ๏Œป๏Œบร—๏Œป we can use this to rewrite each of our expressions as ๐‘Ž๐‘ร—๐‘๐‘Ž.๏Šจร—๏Šช๏Šซร—๏Šช๏Šง๏Šฆร—๏Šจ๏Šชร—๏Šจ Simplifying the exponents gives us ๐‘Ž๐‘ร—๐‘๐‘Ž.๏Šฎ๏Šจ๏Šฆ๏Šจ๏Šฆ๏Šฎ At this point, we can simplify the two expressions by cross canceling: ๐‘Ž๐‘ร—๐‘๐‘Ž.๏Šฎ๏Šง๏Šจ๏Šฆ๏Šง๏Šจ๏Šฆ๏Šง๏Šฎ๏Šง Our answer is, therefore, 11ร—11=1.

Key Points

  • When simplifying expressions involving powers of quotients, we use the following rules of powers:
    • Power rule: (๐‘Ž)=๐‘Ž๏Š๏‰๏Šร—๏‰.
    • Product rule: (๐‘Ž๐‘)=๐‘Ž๐‘๏Š๏Š๏Š.
    • Quotient rule: ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘๏Š๏Š๏Š.
  • The general method is usually to apply the quotient rule first, then the product rule if necessary, and then the power rule to the individual terms.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.