5 answers
To solve the equation \( x^2 = 7^8 \) by inspection, we can take the square root of both sides.
The solutions will be: \[ x = \pm \sqrt{7^8} \]
Since \( \sqrt{7^8} = 7^{8/2} = 7^4 \), we find that: \[ x = 7^4 \quad \text{and} \quad x = -7^4 \]
Now, calculating \( 7^4 \): \[ 7^4 = 2401 \]
Thus, the two real solutions are: \[ -7^4 \text{ and } 7^4 \]
Writing them as required: \[ (-7^4, 7^4) \]
So, the final answer is: \[ { (-7^4), (7^4) } \]
To solve the equation
\[ -9 + (x - 3)^2 - 4 = -4, \]
we first simplify the left-hand side:
Add 9 to both sides: \[ (x - 3)^2 - 4 = 5. \]
Add 4 to both sides: \[ (x - 3)^2 = 9. \]
Now, we take the square root of both sides: \[ x - 3 = \pm 3. \]
This results in two cases:
Case 1: \[ x - 3 = 3 \implies x = 6. \]
Case 2: \[ x - 3 = -3 \implies x = 0. \]
Thus, the two solutions for \( x \) are: \[ x = 6 \quad \text{and} \quad x = 0. \]
Writing them in the required format, we have:
\[ {0, 6}. \]
Now, let's check the provided options:
None of the given options match the solutions found.
If we made a mistake interpreting the question, let me know, but the correct pairs from our calculations are {0, 6}.
The given equation is
\[ -9 + \frac{(x - 3)^2}{-4} = -4. \]
First, we can move the -9 to the right-hand side:
\[ \frac{(x - 3)^2}{-4} = -4 + 9. \]
This simplifies to:
\[ \frac{(x - 3)^2}{-4} = 5. \]
Now, we can multiply both sides by -4 to clear the fraction:
\[ (x - 3)^2 = -4 \times 5. \]
This leads to:
\[ (x - 3)^2 = -20. \]
Since the right side is negative, this equation has no real solutions.
Given that the provided options are all pairs of numbers, there are no valid pairs that match the conditions of the equation. Thus, if the requirement is for real solutions, we can conclude that there are no real solutions.
Could you double-check the formulation of the equation? If there’s a mistake, please provide the corrected equation for accurate assistance!
