If x^2 . x^2 is equivalent to x^4 because 2 + 2 = 4, which two expressions are equivalent to x^4?

Steps to Solve

1. Understanding the given condition

We are told that $x^2 \cdot x^2$ is equivalent to $x^4$. This means that we can use properties of exponents to find other expressions that also equal $x^4$.

2. Using properties of exponents

Recall that when multiplying expressions with the same base, you add the exponents: $$ x^a \cdot x^b = x^{a + b} $$

3. Identifying equivalent expressions

We need to find combinations of the provided options that sum to 4 when using exponent addition. For example, we want to find: $$ a + b = 4 $$

4. Evaluating the first option

The first option is: $$ x^{1/4} \cdot x^{1/4} \cdot x^{1/4} \cdot x^{1/4} $$

Calculating the exponent: $$ 1/4 + 1/4 + 1/4 + 1/4 = 1 $$ So this option does not equal $x^4$.

5. Evaluating the second option

The second option is: $$ x^{1/4} \cdot x^{1/4} \cdot x^{1/4} \cdot x^{1/2} \cdot x^{1/2} \cdot x^{1/2} \cdot x^{1/2} $$

Calculating the exponent: $$ 1/4 + 1/4 + 1/4 + 1/2 + 1/2 + 1/2 + 1/2 = 4 $$ This option equals $x^4$.

6. Continuing with the third option

The third option is: $$ x^{1/3} \cdot x^{1/3} \cdot x^{1/3} \cdot x^{1/3} $$

Calculating the exponent results in: $$ 1/3 + 1/3 + 1/3 + 1/3 = 4/3 $$ This does not equal $x^4$.

7. Evaluating the fourth option

The fourth option is: $$ x^{1/2} \cdot x^{1/2} \cdot x^{1/2} \cdot x^{1/2} $$

Calculating the exponent results in: $$ 1/2 + 1/2 + 1/2 + 1/2 = 2 $$ This option does not equal $x^4$.

8. Final evaluations

After checking each option, we find that the combinations that equal $x^4$ are:

1. The second option: $x^{1/4} \cdot x^{1/4} \cdot x^{1/4} \cdot x^{1/2} \cdot x^{1/2} \cdot x^{1/2} \cdot x^{1/2}$
2. The fourth option: $x^{1/2} \cdot x^{1/2} \cdot x^{1/2} \cdot x^{1/2}$