$$ \begin{array}{ll} ext { c) } 4^{3 x-1}=1 & ext { d) } 5^{x}+5^{x-1}=30\end{array} $$

Question

$$ \begin{array}{ll}	ext { c) } 4^{3 x-1}=1 & 	ext { d) } 5^{x}+5^{x-1}=30\end{array} $$

UpStudy AI · Accepted Answer

Let's solve each equation step by step.

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### **c) Solve $$4^{3x - 1} = 1$$:**

**Step 1:** Recognize that any non-zero base raised to the power of 0 is 1.
$$
4^{3x - 1} = 1 \implies 4^{0} = 1
$$

**Step 2:** Set the exponent equal to 0.
$$
3x - 1 = 0
$$

**Step 3:** Solve for $$x$$.
$$
3x = 1 \implies x = \frac{1}{3}
$$

**Answer:** $$x = \frac{1}{3}$$

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### **d) Solve $$5^{x} + 5^{x - 1} = 30$$:**

**Step 1:** Factor out $$5^{x - 1}$$ from both terms.
$$
5^{x} + 5^{x - 1} = 5^{x - 1} \cdot 5 + 5^{x - 1} = 5^{x - 1}(5 + 1) = 5^{x - 1} \cdot 6
$$

**Step 2:** Set the equation equal to 30.
$$
5^{x - 1} \cdot 6 = 30
$$

**Step 3:** Divide both sides by 6 to isolate $$5^{x - 1}$$.
$$
5^{x - 1} = \frac{30}{6} = 5
$$

**Step 4:** Recognize that $$5^1 = 5$$, so set the exponents equal.
$$
x - 1 = 1
$$

**Step 5:** Solve for $$x$$.
$$
x = 1 + 1 = 2
$$

**Answer:** $$x = 2$$

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**Final Answers:**

- **c)** $$x = \frac{1}{3}$$
- **d)** $$x = 2$$
**Solutions:** - **c)** $$x = \frac{1}{3}$$ - **d)** $$x = 2$$

\( \begin{array}{ll}\text { c) } 4^{3 x-1}=1 & \text { d) } 5^{x}+5^{x-1}=30\end{array} \)