Using the properties of integer exponents, match each expression with its equivalent expression.
[tex]\left(-3^{-2}\right)^0[/tex]
[tex]3^3 \cdot 3^1 \cdot 3^2 \cdot 3^{-12}[/tex]
[tex]-3^5 \div-3^8[/tex]
[tex]-3^{-3} \cdot 3^{-3}[/tex]
Answer (1)
( − 3 − 2 ) 0 = 1 because any non-zero number raised to the power of 0 is 1.
3 3 ⋅ 3 1 ⋅ 3 2 ⋅ 3 − 12 = 3 3 + 1 + 2 − 12 = 3 − 6 using the property a m ⋅ a n = a m + n .
− 3 5 ÷ − 3 8 = 3 5 − 8 = 3 − 3 using the property a n a m = a m − n .
− 3 − 3 ⋅ 3 − 3 = − 3 − 6 using the property a m ⋅ a n = a m + n .
Explanation
Problem Analysis We are asked to match expressions using the properties of integer exponents. Let's analyze each expression separately.
Simplifying Expression 1 The first expression is ( − 3 − 2 ) 0 . Any non-zero number raised to the power of 0 is 1. Therefore, ( − 3 − 2 ) 0 = 1 .
Simplifying Expression 2 The second expression is 3 3 ⋅ 3 1 ⋅ 3 2 ⋅ 3 − 12 . When multiplying exponential terms with the same base, we add the exponents: a m ⋅ a n = a m + n . Thus, we have 3 3 + 1 + 2 − 12 = 3 − 6 .
Simplifying Expression 3 The third expression is − 3 5 ÷ − 3 8 . When dividing exponential terms with the same base, we subtract the exponents: a n a m = a m − n . Thus, we have − 3 8 − 3 5 = 3 5 − 8 = 3 − 3 .
Simplifying Expression 4 The fourth expression is − 3 − 3 ⋅ 3 − 3 . This can be written as − ( 3 − 3 ⋅ 3 − 3 ) . Using the property a m ⋅ a n = a m + n , we get − ( 3 − 3 − 3 ) = − 3 − 6 .
Matching the Expressions Now, we match the simplified expressions with their equivalents:
( − 3 − 2 ) 0 = 1
3 3 ⋅ 3 1 ⋅ 3 2 ⋅ 3 − 12 = 3 − 6
− 3 5 ÷ − 3 8 = 3 − 3
− 3 − 3 ⋅ 3 − 3 = − 3 − 6
Examples
Understanding and manipulating exponents is crucial in many fields, such as computer science when dealing with memory sizes (kilobytes, megabytes, gigabytes, etc.) or in physics when dealing with very large or very small numbers (like the size of atoms or the distance to stars). For example, calculating the storage capacity of a hard drive or determining the energy of a photon involves using exponential notation and its properties.
