Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( g = 2 \)
Solution
To solve the equation \( 5^{4g - 8} = 1 \) for \( g \), follow these steps:
### Step 1: Understand the Exponential Equation
Any non-zero number raised to the power of **0** equals **1**. Mathematically, for \( a \neq 0 \):
\[
a^0 = 1
\]
Given the equation:
\[
5^{4g - 8} = 1
\]
We can deduce that the exponent must be **0** because \( 5 \neq 1 \). Therefore:
\[
4g - 8 = 0
\]
### Step 2: Solve for \( g \)
Now, solve the linear equation for \( g \):
\[
4g - 8 = 0
\]
**Add 8 to both sides:**
\[
4g = 8
\]
**Divide both sides by 4:**
\[
g = \frac{8}{4} = 2
\]
### Conclusion
The solution to the equation \( 5^{4g - 8} = 1 \) is:
\[
g = 2
\]
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Simplify this solution Beyond the Answer
To solve the equation \( 5^{4g-8} = 1 \), we can use the property that any non-zero number raised to the power of 0 equals 1. Therefore, we can set the exponent equal to 0: \[ 4g - 8 = 0 \] Now, solving for \( g \): 1. Add 8 to both sides: \( 4g = 8 \) 2. Divide both sides by 4: \( g = 2 \) So the solution to the equation is \( g = 2 \). Now you can confidently substitute back into the original equation to verify if \( 5^{4(2)-8} = 1 \) holds true, which simplifies to \( 5^0 = 1 \). That's a success!
