Explainer: Powers of Monomials

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‘Ž๐‘›)๐‘š=๐‘Ž๐‘›ร—๐‘š.

We can use this rule to simplify expressions such as ๏€น๐‘ฆ3๏…10, which could be written as ๐‘ฆ3ร—10=๐‘ฆ30.

Here, we are going to look at how we simplify a rational expression raised to a power. For example, ๏€พ๐‘ฅ3๐‘ฆ4๐‘ง5๏Š4.

We can expand the expression to get ๐‘ฅ3๐‘ฆ4๐‘ง5ร—๐‘ฅ3๐‘ฆ4๐‘ง5ร—๐‘ฅ3๐‘ฆ4๐‘ง5ร—๐‘ฅ3๐‘ฆ4๐‘ง5, which is then the same as ๏€น๐‘ฅ3๐‘ฆ4๏…4(๐‘ง5)4.

We can now apply the product rule of exponents, (๐‘Ž๐‘)๐‘›=๐‘Ž๐‘›๐‘๐‘›, to rewrite the numerator as follows: ๏€น๐‘ฅ3๏…4๏€น๐‘ฆ4๏…4(๐‘ง5)4.

At this point, we can apply the power rule for exponents to the top and bottom of the expression to get ๐‘ฅ3ร—4๐‘ฆ4ร—4๐‘ง5ร—4, which simplifies to ๐‘ฅ12๐‘ฆ16๐‘ง20.

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‘›=๐‘Ž๐‘›๐‘๐‘›.

Let us now look at a few examples.

Example 1: Simplifying Powers of Rational Expressions

Simplify ๏€พ๐‘ฅ3๐‘ฆ4๏Š4.

Answer

With this question, we start by using the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡๐‘›=๐‘Ž๐‘›๐‘๐‘›, to rewrite the expression as ๏€น๐‘ฅ3๏…4(๐‘ฆ4)4.

Using the power rule of exponents, (๐‘ฅ๐‘Ž)๐‘=๐‘ฅ๐‘Žร—๐‘, we can simplify the expression: the top simplifies to ๐‘ฅ3ร—4=๐‘ฅ12 and the bottom to ๐‘ฆ4ร—4=๐‘ฆ16. Our final answer is, therefore, ๐‘ฅ12๐‘ฆ16.

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

Simplify ๏€พ๐‘ฅ3๐‘ฆ2๐‘ง3๏Š3.

Answer

Our first step in answering this question is to use the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡๐‘›=๐‘Ž๐‘›๐‘๐‘›, to rewrite the expression as follows: ๏€น๐‘ฅ3๐‘ฆ2๏…3(๐‘ง3)3.

If we then recall the power rule of exponents, which tells us that (๐‘ฅ๐‘Ž)๐‘=๐‘ฅ๐‘Žร—๐‘, we can rewrite our expression as ๐‘ฅ3ร—3๐‘ฆ2ร—3๐‘ง3ร—3.

We can now simplify all of the exponents to get ๐‘ฅ9๐‘ฆ6๐‘ง9.

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

Simplify ๏€พ๐‘Ž2๐‘4๐‘5๐‘‘6๏Š3.

Answer

Our first step in answering this question is to use the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡๐‘›=๐‘Ž๐‘›๐‘๐‘›, to rewrite the expression as follows: ๏€น๐‘Ž2๐‘4๏…3(๐‘5๐‘‘6)3.

If we then recall the power rule of exponents, which tells us that (๐‘ฅ๐‘Ž)๐‘=๐‘ฅ๐‘Žร—๐‘, we can rewrite our expression as ๐‘Ž2ร—3๐‘4ร—3๐‘5ร—3๐‘‘6ร—3.

We can now simplify all of the exponents to get ๐‘Ž6๐‘12๐‘15๐‘‘18.

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

Simplify ๏€น๐‘Žรท๐‘7๏…10.

Answer

We begin by rewriting the expression as a quotient: ๏€น๐‘Žรท๐‘7๏…10=๏€ป๐‘Ž๐‘7๏‡10. We can now use the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡๐‘›=๐‘Ž๐‘›๐‘๐‘›, to rewrite the expression as follows: ๏€น๐‘Žรท๐‘7๏…10=๐‘Ž10(๐‘7)10.

Using the power rule for exponents, we can further simplify the denominator of the expression as follows: ๏€น๐‘Žรท๐‘7๏…10=๐‘Ž10๐‘7ร—10=๐‘Ž10๐‘70.

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

Write an equivalent expression to ๏€พ๐‘ฅ7๐‘ฆ5๐‘ง5๏Š3 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‘›=๐‘Ž๐‘›๐‘๐‘›, to rewrite the expression as follows: ๏€น๐‘ฅ7๐‘ฆ5๏…3(๐‘ง5)3.

We can now use the product rule for exponents to rewrite the numerator as follows: ๏€น๐‘ฅ7๏…3๏€น๐‘ฆ5๏…3(๐‘ง5)3.

At this point, we can use the power rule of exponents, which tells us that (๐‘ฅ๐‘Ž)๐‘=๐‘ฅ๐‘Žร—๐‘, to rewrite our expression as follows: ๐‘ฅ7ร—3๐‘ฆ5ร—3๐‘ง5ร—3.

Finally, we can now simplify all of the exponents to get ๐‘ฅ21๐‘ฆ15๐‘ง15.

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‘Ž2๐‘5๏Š4ร—๏€พ๐‘10๐‘Ž4๏Š2.

Answer

With this question, we can start by using the power of a quotient rule, ๏€ป๐‘Ž๐‘๏‡๐‘›=๐‘Ž๐‘›๐‘๐‘›, to rewrite the expression as follows: ๏€น๐‘Ž2๏…4(๐‘5)4ร—๏€น๐‘10๏…2(๐‘Ž4)2. If we then recall the power rule of exponents, which tells us that (๐‘ฅ๐‘Ž)๐‘=๐‘ฅ๐‘Žร—๐‘, we can use this to rewrite each of our expressions as ๐‘Ž2ร—4๐‘5ร—4ร—๐‘10ร—2๐‘Ž4ร—2. Simplifying the exponents gives us ๐‘Ž8๐‘20ร—๐‘20๐‘Ž8. At this point, we can simplify the two expressions by cross canceling: ๐‘Ž81๐‘201ร—๐‘201๐‘Ž81. Our answer is, therefore, 11ร—11=1.

Key Points

  1. When simplifying expressions involving powers of quotients, we use the following rules of powers:
    1. Power rule: (๐‘Ž๐‘›)๐‘š=๐‘Ž๐‘›ร—๐‘š.
    2. Product rule: (๐‘Ž๐‘)๐‘›=๐‘Ž๐‘›๐‘๐‘›.
    3. Quotient rule: ๏€ป๐‘Ž๐‘๏‡๐‘›=๐‘Ž๐‘›๐‘๐‘›.
  2. 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.

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