Identify equivalent expressions involving exponents

Key Notes :

Understanding Equivalent Expressions

  • Equivalent expressions are expressions that have the same value for all possible values of the variables involved.
  • In the context of exponents, equivalent expressions are expressions that can be simplified to the same form using the properties of exponents.

Key Properties of Exponents

  • Product Rule: a^m * a^n = a^(m+n)
  • Quotient Rule: a^m / a^n = a^(m-n)
  • Power Rule: (a^m)^n = a^(m*n)
  • Zero Exponent Rule: a^0 = 1 (where a ≠ 0)
  • Negative Exponent Rule: a^(-n) = 1/a^n (where a ≠ 0)

Identifying Equivalent Expressions

  1. Apply the properties of exponents to simplify both expressions.
  2. Compare the simplified forms.
  3. If the simplified forms are identical, the expressions are equivalent.

Examples:

  • Are 2³ * 2⁴ and 2⁷ equivalent?
    • Simplify the first expression: 2^(3+4) = 2⁷
    • The simplified forms are identical, so the expressions are equivalent.
  • Are (3²)³ and 3⁶ equivalent?
    • Simplify the first expression: 3^(2*3) = 3⁶
    • The simplified forms are identical, so the expressions are equivalent.

Learn with an example

Select all the expressions that are equivalent to 103 . 10-8

  • 1 / 10-24
  • 10-5
  • 1 / 105
  • 1 / 10-5

To find which expressions are equivalent to 103 .10-8 , write each expression with the same base.

The answer choices and 103 .10-8 can all be written as a power of 10. So, use properties of exponents to write each expression in the form 10 . Then, see which ones match 103 .10-8 in this form.

Step 1: Write 103 .10-8 in the form 10

103 .10-8

103+ -8 Use the identity ax ay = a x+y

10-5

So, 103 .10-8  is equal to 10-5.

Step 2: Now write each answer choice in the form 10

Write 1 / 10-24 in the form  10

1 / 10-24

10-(-24) Use the identity 1/ax = a-x

1024

So,1 / 10-24 is equal to 1024 , but 103 .10-8 is not. (It is equal to 10-5)

The expressions 1 / 10-24 and 103 .10-8 are not equivalent.

The expression  10-5 is in the form 10 . You’ve already found that 103 .10-8  is equal to 10-5

The expressions 10-5 and 103 .10-8 are equivalent.

Write 1 / 105 in the form 10

1 / 105

=10-5 Use the identity 1 / ax = a-x

Both 1 / 105 and 103 .10-8 are equal to 10-5

The expressions 1 / 105 and  103 .10-8 are equivalent.

Write  1 / 10-5 in the form 10

1 / 10-5

= 10-(-5) Use the identity 1 / ax = a-x

= 105

So, 1 / 10-5 is equal to 105 , but 103 .10-8 is not. (It is equal to 10-5)

The expressions 1 / 10-5 and  103 .10-8 are not equivalent.

The correct answers are:

  • 10-5
  • 1 / 105

Select all the expressions that are equivalent to 57 . 5-2

  • 1 / 5-5
  • 1 / 55
  • 1 / 5-14
  • 5-14

To find which expressions are equivalent to 57 . 5-2 , write each expression with the same base.

The answer choices and 57 . 5-2 can all be written as a power of 5. So, use properties of exponents to write each expression in the form 5 . Then, see which ones match 57 . 5-2 in this form.

Step 1: Write 57 . 5-2 in the form 5

57 . 5-2

57+ -2 Use the identity ax ay = a x+y

55

So, 57 . 5-2  is equal to 55.

Step 2: Now write each answer choice in the form 5

Write 1 / 5-5 in the form  5

1 / 5-5

5-(-5) Use the identity 1/ax = a-x

55

Both 1 / 5-5 and 57 . 5-2 are equal to 55

The expressions 1 / 5-5 and 57 . 5-2 are equivalent.

Write 1 / 55 in the form 5

1 / 55

=5-5 Use the identity 1 / ax = a-x

So,1 / 55 is equal to 5-5 , but  57 . 5-2 is not. (It is equal to55)

The expressions 1 / 55 and  57 . 5-2 are not equivalent.

Write  1 / 5-14 in the form 5

1 / 5-14

= 5-(-14) Use the identity 1 / ax = a-x

= 514

So, 1 / 5-14 is equal to 514 , but 57 . 5-2 is not. (It is equal to 55 )

The expressions 1 / 5-14 and  57 . 5-2are not equivalent.

The expression  5-14 is in the form 5 . You’ve already found that 57 . 5-2 is not equal to 5-14(It is equal to 55.)

The expressions 5-14 and 57 . 5-2 are not equivalent.

The correct answers is 1 / 5-5 .

let’s practice!