Question
2.2 Solve for \( x \) WITHOUT using a calculator: \( \begin{array}{ll}2.2 .1 & 2 x^{\frac{3}{4}}=16 \\ 2.2 .2 & 5^{x+1}+5^{x}=150\end{array} \)
Ask by Luna Henry. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**Solutions:**
- **Problem 2.2.1:** \( x = 16 \)
- **Problem 2.2.2:** \( x = 2 \)
Solution
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:** \( x = 2 \)
---
**Final Answers:**
- **Problem 2.2.1:** \( x = 16 \)
- **Problem 2.2.2:** \( x = 2 \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To solve \(2 x^{\frac{3}{4}}=16\), start by dividing both sides by 2, giving you \(x^{\frac{3}{4}}=8\). Next, raise both sides to the power of \(\frac{4}{3}\) to isolate \(x\): \(x = 8^{\frac{4}{3}} = (2^3)^{\frac{4}{3}} = 2^{4} = 16\). So, \(x = 16\). For \(5^{x+1}+5^{x}=150\), notice that \(5^{x+1} = 5 \cdot 5^{x}\), allowing you to rewrite the equation as \(5 \cdot 5^{x} + 5^{x} = 150\), or \((5 + 1)5^{x} = 150\). Thus, \(6 \cdot 5^{x} = 150\). Dividing by 6 gives \(5^{x} = 25\), which implies \(5^x = 5^2\). Hence, \(x = 2\). So, the solutions are \(x = 16\) for the first equation and \(x = 2\) for the second.
