Question

Which functions are equivalent to f(x)=4√162 ? Check all that apply.

a. f(x)=2√2 ⋅ √18

b. f(x)=3√2 ⋅ √9

c. f(x)=6√2 ⋅ √3

d. f(x)=√2 ⋅ √81

Answer 1

After simplifying the square root of 162 and multiplying by 4, we find that it is equivalent to 36√2. Comparing with the given options, only option (c) f(x)=6√2 ⋅ √3 simplifies to the same result, which is 36√2, making it equivalent to f(x)=4√162.

Step-by-step explanation:

The student's question is about determining which functions are equivalent to f(x)=4√162. To find equivalent expressions, we need to simplify and manipulate the given square root using properties of exponents and roots. Remember that when two square roots are multiplied, we can combine them under a single square root if both radicands are non-negative. Also, square roots can be expressed as fractional exponents such that x² = √x.

First let's simplify √162:

• √162 = √(81*2) = √81 * √2 = 9√2

Now we multiply by 4:

• 4 * 9√2 = 36√2

Let's use the same approach to simplify and compare the given options:

1. f(x)=2√2 ⋅ √18 = 2√2 * 3√2 = 6√4 = 6(2) = 12
2. f(x)=3√2 ⋅ √9 = 3√2 * 3 = 9√2
3. f(x)=6√2 ⋅ √3 = 6√2 * √3 = 6√6 = 36√2
4. f(x)=√2 ⋅ √81 = √2 * 9 = 9√2

From the above analysis, we find that only option (c) is equivalent to f(x)=4√162.