Part A: If (7^2)^x = 1, what is the value of x? Explain your answer. (5 points)
Part B: If (7^0)^x = 1, what are the possible values of x? Explain your answer. (5 points)
Part A: If (7^2)^x = 1, we can simplify the expression by applying the exponent rule which states that (a^b)^c = a^(b*c). Therefore, we have 7^(2*x) = 1.
In order for an exponentiation to result in 1, the base must be 1, since any non-zero number raised to the power of 0 is equal to 1.
Therefore, we have 7^(2*x) = 1, which means 7^(2*x) = 7^0. By equating the exponents, we can conclude that 2*x = 0.
Solving for x, we divide both sides of the equation by 2, yielding x = 0/2, which simplifies to x = 0.
So the value of x in the equation is 0.
Part B: If (7^0)^x = 1, we can simplify the expression by again applying the exponent rule, which states that a^0 = 1, where a is any non-zero number.
Therefore, we have 1^x = 1.
Since any number raised to the power of 1 is itself, we can conclude that x must be any real number.
Therefore, the possible values of x are all real numbers.
Part A:
To find the value of x in the equation (7^2)^x = 1, we need to simplify the equation step by step.
Step 1: Simplify the expression inside the parentheses: 7^2 = 49.
So, the equation becomes: 49^x = 1.
Step 2: Recall that any number raised to the power of 0 is equal to 1.
So, 49^0 = 1.
Step 3: Apply the rule of exponents that states when raising a power to a power, you multiply the exponents. In this case, the exponent of x is multiplying the exponent of 49.
Since we know that 49^0 = 1, we get 1^x = 1.
Step 4: Any number raised to the power of x, where x is any real number, is always equal to 1.
Therefore, x can take any real value.
Answer: The value of x in the equation (7^2)^x = 1 is any real number.
Part B:
In the equation (7^0)^x = 1, we need to find the possible values of x.
Step 1: Recall that any number (except 0) raised to the power of 0 is always equal to 1.
So, 7^0 = 1.
Step 2: Apply the rule of exponents that states when raising a power to a power, you multiply the exponents. In this case, the exponent of x is multiplying the exponent of 1.
Since we know that 7^0 = 1, we get 1^x = 1.
Step 3: Any number raised to the power of x, where x is any real number, is always equal to 1.
Therefore, x can take any real value.
Answer: The possible values of x in the equation (7^0)^x = 1 are any real numbers.
Part A: To solve the equation (7^2)^x = 1, we need to understand the properties of exponents. In this case, we have a power of a power, so we can apply the exponent property, which states that (a^m)^n = a^(m*n).
For our equation, we have (7^2)^x = 1. Applying the exponent property, we can rewrite it as 7^(2*x) = 1.
Next, we need to recall that any number raised to the power of 0 equals 1. Therefore, if 7^(2*x) = 1, then 2*x must equal 0.
To solve for x, we divide both sides of the equation 2*x = 0 by 2, resulting in x = 0/2, which simplifies to x = 0.
Therefore, the value of x in the equation (7^2)^x = 1 is 0.
Part B: Similarly, for the equation (7^0)^x = 1, we apply the exponent property that any number raised to the power of 0 equals 1.
However, this equation differs from Part A because we have an x in the exponent. To find the possible values of x, we need to consider the cases when the equation holds true.
If 7^0 equals 1, then for any value of x, (7^0)^x will still equal 1. This means that any real number is a possible value for x.
Therefore, the possible values for x in the equation (7^0)^x = 1 are all real numbers.