Which expressions are equivalent to $\sqrt[3]{128}^x$ ? Select three correct answers.
$128^{\frac{x}{3}}$
$128^{\frac{3}{x}}$
$(4 \sqrt[3]{2})^x$
$\left(4\left(2^{\frac{1}{3}}\right)\right)^x$
$(2 \sqrt[3]{4})^x$
Answer (2)
12 8 3 x , ( 4 3 2 ) x , ( 4 ( 2 3 1 ) ) x
Explanation
Understanding the Problem We are given the expression 3 128 x and asked to find three equivalent expressions from the list: 12 8 3 x , 12 8 x 3 , ( 4 3 2 ) x , ( 4 ( 2 3 1 ) ) x , ( 2 3 4 ) x .
Simplifying the Original Expression First, let's rewrite the given expression using exponent rules: 3 128 x = ( 12 8 3 1 ) x = 12 8 3 x . So, the first option 12 8 3 x is equivalent.
Expressing with Base 2 Now, let's simplify 128 . Since 128 = 2 7 , we can rewrite the expression as: 12 8 3 x = ( 2 7 ) 3 x = 2 3 7 x .
Simplifying Option 3 Next, let's examine the third option, ( 4 3 2 ) x . We can rewrite this as ( 2 2 ⋅ 2 3 1 ) x = ( 2 2 + 3 1 ) x = ( 2 3 7 ) x = 2 3 7 x . Thus, ( 4 3 2 ) x is equivalent to the original expression.
Simplifying Option 4 Now, let's examine the fourth option, ( 4 ( 2 3 1 ) ) x . We can rewrite this as ( 4 ⋅ 2 3 1 ) x = ( 2 2 ⋅ 2 3 1 ) x = ( 2 2 + 3 1 ) x = ( 2 3 7 ) x = 2 3 7 x . Thus, ( 4 ( 2 3 1 ) ) x is equivalent to the original expression.
Simplifying Option 5 Finally, let's examine the fifth option, ( 2 3 4 ) x . We can rewrite this as ( 2 ⋅ 4 3 1 ) x = ( 2 ⋅ ( 2 2 ) 3 1 ) x = ( 2 ⋅ 2 3 2 ) x = ( 2 1 + 3 2 ) x = ( 2 3 5 ) x = 2 3 5 x . This is not equivalent to the original expression.
Final Answer Therefore, the three expressions equivalent to 3 128 x are 12 8 3 x , ( 4 3 2 ) x , and ( 4 ( 2 3 1 ) ) x .
Examples
Understanding exponential expressions and their equivalent forms is crucial in many scientific and engineering fields. For example, in finance, compound interest calculations often involve exponential growth. Similarly, in physics, radioactive decay is modeled using exponential decay functions. Being able to manipulate and simplify these expressions allows for easier analysis and prediction of these phenomena.
The expressions equivalent to 3 128 x are 12 8 3 x , ( 4 3 2 ) x , and ( 4 ( 2 3 1 ) ) x .
;
