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Lesson Explainer: Simplifying Exponential Expressions with Integer Exponents Mathematics

In this explainer, we will learn how to perform operations and simplifications on expressions that involve integer exponents.

To help us understand how to simplify expressions involving integer exponents, we will recall how to evaluate a power. Let’s first recall how to evaluate a power with a positive integer exponent.

Definition: Evaluating a Power

For a power with base π‘Ž where π‘Žβˆˆβ„βˆ’0 and exponent 𝑛 where π‘›βˆˆβ„€οŠ°, then π‘Ž=π‘ŽΓ—π‘ŽΓ—π‘ŽΓ—β‹―Γ—π‘Ž.ο‡Œο†²ο†²ο†²ο†²ο†²ο‡ο†²ο†²ο†²ο†²ο†²ο‡ŽοŠοŠtimes

Next, we will recall how to evaluate a power with a zero or negative base.

Laws: Rules for Zero and Negative Exponents

  • The law for zero exponents: π‘Ž=1, where π‘Žβˆˆβ„βˆ’0
  • The law for negative exponents: π‘Ž=1π‘Žπ‘Ž=1π‘Ž,or where π‘Žβˆˆβ„βˆ’0 and π‘›βˆˆβ„€

Using this definition along with the order of operations, we can simplify numerical and algebraic expressions with integer exponents. We will do this in our first example.

Example 1: Evaluating an Expression Involving Integer Exponents

Calculate 10Γ—25Γ—420Γ—5.

Answer

To calculate 10Γ—25Γ—420Γ—5, we can write the powers in expanded form and then cancel common factors in the numerator and denominator.

Recall that the expanded form is when we write a power with base π‘Ž where π‘Žβˆˆβ„βˆ’0 and exponent 𝑛 where π‘›βˆˆβ„€οŠ° as π‘Ž=π‘ŽΓ—π‘ŽΓ—π‘ŽΓ—β‹―Γ—π‘Ž.

In our example, this would give us 10Γ—25Γ—420Γ—5=10Γ—10Γ—10Γ—10Γ—10Γ—25Γ—25Γ—420Γ—20Γ—5Γ—5Γ—5Γ—5Γ—5.

We can cancel common factors by writing the numerator and denominator as products of prime factors first and then canceling. Doing so gives us 10Γ—25Γ—420Γ—5=10Γ—10Γ—10Γ—10Γ—10Γ—25Γ—25Γ—420Γ—20Γ—5Γ—5Γ—5Γ—5Γ—5=ο€Ό2Γ—5οˆΓ—ο€Ό2Γ—5οˆΓ—ο€Ό2Γ—5οˆΓ—ο€Ό2Γ—5οˆΓ—ο€Ό2Γ—5οˆΓ—ο€Ό5Γ—5οˆΓ—ο€Ί5Γ—5×2Γ—22Γ—2Γ—5οˆΓ—ο€Ό2Γ—2Γ—5οˆΓ—5Γ—5Γ—5Γ—5Γ—5=2Γ—5Γ—5Γ—2Γ—21=200.

Therefore, 10Γ—25Γ—420Γ—5=200.

Sometimes, when simplifying expressions with integer exponents, we need to use the laws of exponents to do this. Let’s recall the laws of exponents.

Rule: Laws of Exponents

The following are the rules of exponents with real bases and integer exponents:

  • The product rule: π‘ŽΓ—π‘Ž=π‘Ž,ο‰οŠο‰οŠ°οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€
  • The quotient rule: π‘ŽΓ·π‘Ž=π‘Ž,ο‰οŠο‰οŠ±οŠ where π‘Žβˆˆβ„βˆ’0, π‘šβˆˆβ„€, and π‘›βˆˆβ„€
  • The power of a product rule: (π‘Žπ‘)=π‘Žπ‘, where π‘Ž, π‘βˆˆβ„βˆ’0 and π‘›βˆˆβ„€
  • The power of a quotient rule: ο€»π‘Žπ‘ο‡=π‘Žπ‘, where π‘Ž, π‘βˆˆβ„βˆ’0 and π‘›βˆˆβ„€
  • The power rule: (π‘Ž)=π‘Ž,ο‰οŠο‰οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€

In our next example, we will use the rules of exponents to simplify an expression with square roots. Recall that for π‘Žβ‰₯0, ο€Ίβˆšπ‘Žο†=π‘ŽοŠ¨. We will also use this in the next example.

Example 2: Simplifying Numerical Expressions Involving Square Roots Using Laws of Exponents

Simplify ο€»βˆš3ο‡Γ—ο€»βˆ’βˆš3ο‡ο€»βˆš3ο‡οŠ¨οŠ§οŠ©οŠ©.

Answer

To simplify the expression ο€»βˆš3ο‡Γ—ο€»βˆ’βˆš3ο‡ο€»βˆš3ο‡οŠ¨οŠ§οŠ©οŠ©, we can use laws of exponents. We will start by simplifying the numerator ο€»βˆš3ο‡Γ—ο€»βˆ’βˆš3ο‡οŠ¨οŠ§οŠ©.

As the bases are different, it is helpful to write ο€»βˆ’βˆš3ο‡οŠ§οŠ© as ο€»βˆ’1Γ—βˆš3ο‡οŠ§οŠ© and use the power of a product rule, which states (π‘Žπ‘)=π‘Žπ‘, where π‘Ž, π‘βˆˆβ„βˆ’0 and π‘›βˆˆβ„€. For ο€»βˆ’1Γ—βˆš3ο‡οŠ§οŠ©, this gives us ο€»βˆ’1Γ—βˆš3=(βˆ’1)ο€»βˆš3.

Substituting this back into the numerator, we get ο€»βˆš3(βˆ’1)ο€»βˆš3ο‡οŠ¨οŠ§οŠ©οŠ§οŠ©.

For the powers with base √3, we can then use the product rule, which states π‘ŽΓ—π‘Ž=π‘Ž,ο‰οŠο‰οŠ°οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€. This then gives us ο€»βˆš3(βˆ’1)ο€»βˆš3=(βˆ’1)ο€»βˆš3=(βˆ’1)ο€»βˆš3.

Since 15 is an odd exponent, then (βˆ’1)=βˆ’1, giving us (βˆ’1)ο€»βˆš3=βˆ’1Γ—ο€»βˆš3=βˆ’ο€»βˆš3.

Substituting this back into the expression, we get ο€»βˆš3ο‡Γ—ο€»βˆ’βˆš3ο‡ο€»βˆš3=βˆ’ο€»βˆš3ο‡ο€»βˆš3.

Next, we can use the quotient rule for exponents, which states π‘ŽΓ·π‘Ž=π‘Ž,ο‰οŠο‰οŠ±οŠ where π‘Žβˆˆβ„βˆ’0, π‘šβˆˆβ„€, and π‘›βˆˆβ„€.

Applying this, we get βˆ’ο€»βˆš3ο‡ο€»βˆš3=βˆ’ο€»βˆš3ο‡Γ·ο€»βˆš3=βˆ’ο€»βˆš3=βˆ’ο€»βˆš3.

To simplify βˆ’ο€»βˆš3ο‡οŠ§οŠ¨, we can write this as a product of squares and then use the rule π‘Ž>0,ο€Ίβˆšπ‘Žο†=π‘Ž.

This then gives us βˆ’ο€»βˆš3=βˆ’ο€»βˆš3ο‡Γ—ο€»βˆš3ο‡Γ—ο€»βˆš3ο‡Γ—ο€»βˆš3ο‡Γ—ο€»βˆš3ο‡Γ—ο€»βˆš3=βˆ’3Γ—3Γ—3Γ—3Γ—3Γ—3=βˆ’729.

Therefore, ο€»βˆš3ο‡Γ—ο€»βˆ’βˆš3ο‡ο€»βˆš3ο‡οŠ¨οŠ§οŠ©οŠ© simplified is βˆ’729.

Note that we can also evaluate βˆ’ο€»βˆš3ο‡οŠ§οŠ¨ using the power rule as follows: βˆ’ο€»βˆš3=βˆ’ο€»βˆš3=βˆ’ο€½ο€»βˆš3=βˆ’(3)=βˆ’729.οŠ§οŠ¨οŠ¬Γ—οŠ¨οŠ¨οŠ¬οŠ¬

In the next example, we will consider how to simplify algebraic expressions using the laws of exponents.

Example 3: Using Laws of Exponents to Evaluate Algebraic Expressions

Simplify 4Γ—4(4)Γ—16οŠͺο—οŠ©ο—οŠ±οŠͺ.

Answer

To simplify the expression 4Γ—4(4)Γ—16οŠͺο—οŠ©ο—οŠ±οŠͺ, we need to use the laws of exponents.

First, we will start by simplifying the numerator, 4Γ—4οŠͺ. To do so, we use the product rule for exponents, which states that π‘ŽΓ—π‘Ž=π‘Ž,ο‰οŠο‰οŠ°οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€. This gives us 4Γ—4=4=4.οŠͺο—ο—οŠ°οŠͺο—οŠ«ο—

Next, we will simplify the denominator, (4)Γ—16οŠ©ο—οŠ±οŠͺ. In order to apply the rules for exponents, we need the bases to be the same. As such, we want to write 16 with a base of 4. Since 16 is 4, then we can write 16=ο€Ή4.ο—οŠ¨ο—

To simplify this further, we can use the power law for exponents, which states that (π‘Ž)=π‘Ž,ο‰οŠο‰οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€. This then gives us ο€Ή4=4=4.οŠ¨ο—οŠ¨Γ—ο—οŠ¨ο—

Substituting 4οŠ¨ο— back into the denominator, we now have (4)Γ—16=(4)Γ—4.οŠ©ο—οŠ±οŠͺο—οŠ©ο—οŠ±οŠͺοŠ¨ο—

Since the bases are now the same, we can use the product rule for exponents again to simplify the denominator. This gives us (4)Γ—4=4=4.οŠ©ο—οŠ±οŠͺοŠ¨ο—οŠ©ο—οŠ±οŠͺοŠ°οŠ¨ο—οŠ«ο—οŠ±οŠͺ

Substituting 4οŠ«ο— into the numerator and 4οŠ«ο—οŠ±οŠͺ into the denominator of our original expression gives us 4Γ—4(4)Γ—16=44.οŠͺο—οŠ©ο—οŠ±οŠͺο—οŠ«ο—οŠ«ο—οŠ±οŠͺ

As we have a division, we can use the quotient rule for exponents to simplify the expression further. This rule states that π‘ŽΓ·π‘Ž=π‘Ž,ο‰οŠο‰οŠ±οŠ where π‘Žβˆˆβ„βˆ’0, π‘šβˆˆβ„€, and π‘›βˆˆβ„€. This then gives us 44=4=4=4.οŠ«ο—οŠ«ο—οŠ±οŠͺοŠ«ο—οŠ±(οŠ«ο—οŠ±οŠͺ)οŠ«ο—οŠ±οŠ«ο—οŠ°οŠͺοŠͺ

Evaluating 4οŠͺ, we get 4Γ—4Γ—4Γ—4=256. Therefore, 4π‘₯Γ—4(4)Γ—16=256.οŠͺο—οŠ©ο—οŠ±οŠͺ

Next, we will look at an example where the laws of exponents are used along with common algebraic methods to simplify.

Example 4: Simplifying Rational Algebraic Expressions Using Laws of Exponents

Simplify 6Γ—6Γ—6Γ—6Γ—6Γ—66+6+6+6+6+6.

Answer

To simplify 6Γ—6Γ—6Γ—6Γ—6Γ—66+6+6+6+6+6, we need to consider which laws of exponents can be used and when.

Since the numerator of the expression contains products of powers of the same base, then we can use the product rule. This rule states π‘ŽΓ—π‘Ž=π‘Ž,ο‰οŠο‰οŠ°οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€.

Therefore, for the numerator, we get 6Γ—6Γ—6Γ—6Γ—6Γ—6=6=6.

As the denominator of the expressions contains the addition of powers with the base, then we need to be careful not to incorrectly apply the product rule. Instead, we can gather like terms by adding up the number of 6s, giving us 6 lots of 6 as follows: 6+6+6+6+6+6=6(6).

Since we are multiplying 6 by 6, then we can apply the product rule again to get 6(6)=6.

Substituting the numerator and denominator back into the expression, we get 6Γ—6Γ—6Γ—6Γ—6Γ—66+6+6+6+6+6=66.

As both the numerator and denominator have the same base, then we can apply the quotient rule for exponents. This rule states π‘ŽΓ·π‘Ž=π‘Ž,ο‰οŠο‰οŠ±οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€.

Therefore, applying this, we get 66=6÷6=6=6=6.()

Therefore, 6Γ—6Γ—6Γ—6Γ—6Γ—66+6+6+6+6+6=6.

In the following example, we will use the laws of exponents to simplify algebraic expressions with two different bases.

Example 5: Simplifying Algebraic Fractions Using Properties of Exponents

Simplify π‘₯𝑦π‘₯𝑦()(οŠͺ)().

Answer

To simplify the expression π‘₯𝑦π‘₯𝑦()(οŠͺ)(), we can use the laws of exponents. Since the laws only apply to exponents with the same bases, then it is helpful to rewrite the expression as a product of two fractions that have numerators and denominators with the same base.

This gives us π‘₯𝑦π‘₯𝑦=π‘₯π‘₯×𝑦𝑦.()(οŠͺ)()()()(οŠͺ)

Taking the first fraction, π‘₯π‘₯()(), we can simplify this using the quotient rule for exponents, which states π‘ŽΓ·π‘Ž=π‘Ž,ο‰οŠο‰οŠ±οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€.

This then gives us π‘₯π‘₯=π‘₯=π‘₯=π‘₯.()()()()

Next, we will use the same law to simplify the second denominator, 𝑦𝑦(οŠͺ), giving us 𝑦𝑦=𝑦=𝑦.(οŠͺ)(οŠͺ)οŠͺ

Substituting π‘₯ and 𝑦οŠͺ back into π‘₯π‘₯×𝑦𝑦()()(οŠͺ), we get π‘₯π‘₯×𝑦𝑦=π‘₯×𝑦=π‘₯𝑦.()()(οŠͺ)οŠͺοŠͺ

In the last example, we will discuss how to simplify an algebraic expression using laws of exponents and by taking out common factors that are powers.

Example 6: Using Laws of Exponents to Find an Unknown in a Given Equation

Find the value of π‘Ž for which 2βˆ’2=π‘ŽΓ—2ο—οŠ°οŠ¬ο—οŠ°οŠ¨ο—.

Answer

To find the value of π‘Ž in the equation 2βˆ’2=π‘ŽΓ—2ο—οŠ°οŠ¬ο—οŠ°οŠ¨ο—, we need the left-hand side of the equation to be in the form of the right-hand side. In other words, we want it to be in the form of 𝑏×2=π‘ŽΓ—2, meaning 𝑏=π‘Ž. To do this, we must first manipulate the powers on the left-hand side so that they are written as 2.

For 2ο—οŠ°οŠ¬, we can use the product rule for exponents, which states π‘ŽΓ—π‘Ž=π‘Ž,ο‰οŠο‰οŠ°οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€. This then gives us 2=2Γ—2=64Γ—2,ο—οŠ°οŠ¬ο—οŠ¬ο— which is in the form required.

For 2ο—οŠ°οŠ¨, we can use the product rule again, giving us 2=2Γ—2=2Γ—4=4Γ—2,ο—οŠ°οŠ¨ο—οŠ¨ο—ο— which is in the form required.

Substituting 64Γ—2 and 4Γ—2 in the left-hand side of the equation, we get 64Γ—2βˆ’4Γ—2=π‘ŽΓ—2.

Subtracting 4Γ—2 from 64Γ—2 then gives us 60Γ—2=π‘ŽΓ—2.

Comparing the coefficients of each of the terms, then we can see that π‘Ž=60.

In this explainer, we have learned how to simplify and evaluate expressions with integer exponents. Let’s recap the key points.

Key Points

  • We can simplify exponential expressions by calculating the value of powers and canceling down common factors.
  • We can simplify exponential expressions using the laws of exponents, which are as follows:
    • The product rule for exponents: π‘ŽΓ—π‘Ž=π‘Ž,ο‰οŠο‰οŠ°οŠ where π‘Žβˆˆβ„βˆ’0 and π‘š, π‘›βˆˆβ„€
    • The quotient rule for exponents: π‘ŽΓ·π‘Ž=π‘Ž,ο‰οŠο‰οŠ±οŠ where π‘Žβˆˆβ„βˆ’0, π‘šβˆˆβ„€, and π‘›βˆˆβ„€
    • The power of a product rule: (π‘Žπ‘)=π‘Žπ‘, where π‘Ž, π‘βˆˆβ„βˆ’0 and π‘›βˆˆβ„€
    • The power of a quotient rule: ο€»π‘Žπ‘ο‡=π‘Žπ‘, where π‘Ž, π‘βˆˆβ„βˆ’0 and π‘›βˆˆβ„€
    • The power rule for exponents: (π‘Ž)=π‘Ž,ο‰οŠο‰οŠ where π‘Žβˆˆβ„ and π‘š, π‘›βˆˆβ„€
  • We can use the laws of exponents to solve equations.

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