2.2 Solve for $$ x $$ WITHOUT using a calculator: $$ \begin{array}{ll}2.2 .1 & 2 x^{\frac{3}{4}}=16 \ 2.2 .2 & 5^{x+1}+5^{x}=150\end{array} $$

Question

UpStudy AI · Accepted Answer

Let's solve each equation step by step without using a calculator.

### Problem 2.2.1: $$ 2x^{\frac{3}{4}} = 16 $$

**Step 1:** Isolate $$ x^{\frac{3}{4}} $$

$$
x^{\frac{3}{4}} = \frac{16}{2} = 8
$$

**Step 2:** Solve for $$ x $$

To solve for $$ x $$, we can raise both sides of the equation to the power of $$ \frac{4}{3} $$:

$$
x = 8^{\frac{4}{3}}
$$

**Step 3:** Simplify $$ 8^{\frac{4}{3}} $$

First, find the cube root of 8:

$$
8^{\frac{1}{3}} = 2
$$

Then, raise 2 to the 4th power:

$$
2^4 = 16
$$

**Solution:** $$ x = 16 $$

### Problem 2.2.2: $$ 5^{x+1} + 5^x = 150 $$

**Step 1:** Factor out $$ 5^x $$

$$
5^x (5 + 1) = 150
$$

$$
5^x \times 6 = 150
$$

**Step 2:** Isolate $$ 5^x $$

$$
5^x = \frac{150}{6} = 25
$$

**Step 3:** Solve for $$ x $$

Since $$ 25 = 5^2 $$, we have:

$$
5^x = 5^2
$$

Therefore,

$$
x = 2
$$

**Solution:** $$ x = 2 $$

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**Final Answers:**

- **Problem 2.2.1:** $$ x = 16 $$
- **Problem 2.2.2:** $$ x = 2 $$
**Solutions:** - **Problem 2.2.1:** $$ x = 16 $$ - **Problem 2.2.2:** $$ x = 2 $$

2.2 Solve for \( x \) WITHOUT using a calculator: \( \begin{array}{ll}2.2 .1 & 2 x^{\frac{3}{4}}=16 \\ 2.2 .2 & 5^{x+1}+5^{x}=150\end{array} \)