Examine and complete the work simplifying the expression $\left(2^3\right)^3$.
1. Expand using 3 factors of $2^3$:
2. Apply the product of a power:
$\left(2^3\right)\left(2^3\right)\left(2^3\right)$
$2^{3+3+3}$
3. Simplify:
$2^x$
Answer the questions about simplifying and evaluating the expression $\left(2^3\right)^3$.
What is the value of $x$ in the simplified power?
$\square$
What is the value of the expression?
$\square$
Answer (1)
Simplify the expression using the power of a power rule: ( a m ) n = a m × n , so ( 2 3 ) 3 = 2 3 × 3 = 2 9 .
Identify the value of x in the simplified power 2 x : x = 9 .
Calculate the value of the expression 2 9 : 2 9 = 512 .
The value of x is 9, and the value of the expression is 512 .
Explanation
Understanding the problem We are asked to simplify the expression ( 2 3 ) 3 and find the value of x in the simplified power 2 x , and the value of the expression. Let's break it down step by step.
Applying the power of a power rule First, we expand the expression using the rule that ( a m ) n = a m × n . In our case, we have ( 2 3 ) 3 = 2 3 × 3 = 2 9 .
Finding the value of x Now, we need to find the value of x in the simplified power 2 x . From the previous step, we have 2 9 , so x = 9 .
Calculating the value of the expression Next, we need to calculate the value of the expression 2 9 . This means we need to multiply 2 by itself 9 times: 2 9 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512 .
Final Answer Therefore, the value of x is 9, and the value of the expression ( 2 3 ) 3 is 512.
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with memory sizes (e.g., kilobytes, megabytes, gigabytes, which are powers of 2). For instance, if a file is 2 10 bytes, it means the file size is 1024 bytes. Similarly, in finance, compound interest calculations involve exponents. If you invest $1000 at an annual interest rate of 5%, the amount after 10 years is 1000 × ( 1.05 ) 10 , where the exponent represents the number of years the money is invested.
