Which expressions, when simplified, have a value of $16 ?$ (Select all that apply.)
[tex]2 \sqrt{64}[/tex] [tex]8^2[/tex] [tex]2 \sqrt{8}[/tex] [tex]2^4[/tex]
Answer (1)
2 64 , 2 4
Explanation
Problem Analysis We are given four expressions and asked to identify which simplify to 16. Let's analyze each expression individually.
Simplifying Expression 1
2 64 : The square root of 64 is 8, so this expression becomes 2 × 8 = 16 .
Simplifying Expression 2
8 2 : This means 8 multiplied by itself, which is 8 × 8 = 64 .
Simplifying Expression 3
2 8 : We can simplify 8 as 4 × 2 = 2 2 . Thus, the expression becomes 2 × 2 2 = 4 2 , which is approximately 4 × 1.414 = 5.656 .
Simplifying Expression 4
2 4 : This means 2 multiplied by itself four times, which is 2 × 2 × 2 × 2 = 16 .
Identifying the Correct Expressions Comparing the simplified values to 16, we find that 2 64 and 2 4 both equal 16.
Final Answer Therefore, the expressions that simplify to 16 are 2 64 and 2 4 .
Examples
Understanding how to simplify expressions with square roots and exponents is useful in many areas, such as calculating areas and volumes. For example, if you are designing a square garden with an area of 64 square feet, you need to know that each side is 64 = 8 feet long. If you then want to build a fence around the garden, you'll need to calculate the perimeter, which involves multiplying the side length by 4. Similarly, understanding exponents helps in calculating compound interest or population growth.
