Question

Determine whether each equation is true for all values of x.

1.) 7^3x = 3^7x

A) Yes
B) No

2.) 64^x = 4^3x

A) Yes
B) No

3.) 2^4x = 16^x

A) Yes
B) No

4.) 6^5x = 30x

A) Yes
B) No

Answer 1

Upon applying the properties of exponents to the given equations, we find that equations 2 and 3 are true for all values of x, while equations 1 and 4 are not.

Step-by-step explanation:

We need to determine whether each equation is true for all values of x. For this, we will use the properties of exponents to simplify the equations and see if they hold true for any value of x.

• 1.) 7^3x = 3^7x: This equation is not true for all values of x because the bases are different and no exponent property can equate them. So the answer is B) No.
• 2.) 64^x = 4^3x: If we rewrite 64 as 4^3, we get (4^3)^x = 4^(3x). By the power of a power rule, this simplifies to 4^(3x) = 4^(3x), which is true for all values of x. So the answer is A) Yes.
• 3.) 2^4x = 16^x: Rewriting 16 as 2^4, we get 2^(4x) = (2^4)^x, which simplifies to 2^(4x) = 2^(4x), true for all values of x. So the answer is A) Yes.
• 4.) 6^5x = 30x: Since there's no exponent rule that allows a base of 6 raised to a power to equal a non-exponential function of x, this equation is not always true. So the answer is B) No.

To eliminate terms wherever possible and check the answer to see if it is reasonable are good practices when working with exponents.