Using the properties of integer exponents, match each expression with its equivalent expression.
Tiles:
$4^{-2}$
$4^{-8}$
$4^8$
$4^2$
$4^4$
Pairs:
$\left(4^{-2}\right)^4 \longrightarrow$
$\left(4^2\right)^{-1} \longrightarrow$
$4^2 \cdot 4^6 \longrightarrow$
$4^5 \cdot 4^{-3} \longrightarrow$
Answer (1)
Simplify each expression using exponent rules: ( a m ) n = a m c d o t n and a m c d o t a n = a m + n .
( 4 − 2 ) 4 = 4 − 8 .
( 4 2 ) − 1 = 4 − 2 .
4 2 ⋅ 4 6 = 4 8 .
4 5 ⋅ 4 − 3 = 4 2 . The final matches are: ( 4 − 2 ) 4 ⟶ 4 − 8 , ( 4 2 ) − 1 ⟶ 4 − 2 , 4 2 ⋅ 4 6 ⟶ 4 8 , 4 5 ⋅ 4 − 3 ⟶ 4 2 .
Explanation
Understanding the Problem We are given a list of expressions with integer exponents: 4 − 2 , 4 − 8 , 4 8 , 4 2 , 4 4 . We are also given another list of expressions with integer exponents: ( 4 − 2 ) 4 , ( 4 2 ) − 1 , 4 2 ⋅ 4 6 , 4 5 ⋅ 4 − 3 . Our goal is to match each expression in the second list with its equivalent expression in the first list using the properties of integer exponents.
Reviewing Exponent Properties Let's simplify each expression in the second list using the properties of exponents. Remember, when you raise a power to a power, you multiply the exponents: ( a m ) n = a m ⋅ n . Also, when you multiply powers with the same base, you add the exponents: a m ⋅ a n = a m + n .
Simplifying the First Expression First expression: ( 4 − 2 ) 4 . Multiplying the exponents, we get 4 − 2 ⋅ 4 = 4 − 8 .
Simplifying the Second Expression Second expression: ( 4 2 ) − 1 . Multiplying the exponents, we get 4 2 ⋅ ( − 1 ) = 4 − 2 .
Simplifying the Third Expression Third expression: 4 2 ⋅ 4 6 . Adding the exponents, we get 4 2 + 6 = 4 8 .
Simplifying the Fourth Expression Fourth expression: 4 5 ⋅ 4 − 3 . Adding the exponents, we get 4 5 + ( − 3 ) = 4 5 − 3 = 4 2 .
Matching the Expressions Now, let's match the simplified expressions with the expressions in the first list:
4 − 8 matches 4 − 8
4 − 2 matches 4 − 2
4 8 matches 4 8
4 2 matches 4 2
The expression 4 4 in the first list remains unmatched.
Final Matches Therefore, the matches are:
( 4 − 2 ) 4 ⟶ 4 − 8
( 4 2 ) − 1 ⟶ 4 − 2
4 2 ⋅ 4 6 ⟶ 4 8
4 5 ⋅ 4 − 3 ⟶ 4 2
Examples
Understanding and manipulating exponents is crucial in many fields, such as computer science, where data sizes are often expressed in powers of 2 (e.g., kilobytes, megabytes, gigabytes). For instance, a kilobyte (KB) is 2 10 bytes, a megabyte (MB) is 2 20 bytes, and a gigabyte (GB) is 2 30 bytes. Simplifying expressions with exponents helps in calculating storage capacities and data transfer rates. Also, exponents are used in calculating exponential growth and decay in biology and finance.
