Question

solve the radical equation 3^5sqrt3x^2-10 +7 = 4

Answer 1

To solve the equation \( 3^{\frac{5\sqrt{3x^2 - 10}}{7}} = 4 \), let's first isolate the radical expression.

1. **Isolate the exponential term:**
\[
3^{\frac{5\sqrt{3x^2 - 10}}{7}} = 4
\]
We can rewrite 4 as a power of 3:
\[
4 = 3^{\log_3(4)}
\]

2. **Set the exponents equal:**
Since the bases are the same, we can set the exponents equal to each other:
\[
\frac{5\sqrt{3x^2 - 10}}{7} = \log_3(4)
\]

3. **Multiply both sides by 7:**
\[
5\sqrt{3x^2 - 10} = 7\log_3(4)
\]

4. **Divide by 5:**
\[
\sqrt{3x^2 - 10} = \frac{7\log_3(4)}{5}
\]

5. **Square both sides:**
\[
3x^2 - 10 = \left( \frac{7\log_3(4)}{5} \right)^2
\]

6. **Solve for \( x^2 \):**
\[
3x^2 = \left( \frac{7\log_3(4)}{5} \right)^2 + 10
\]
\[
x^2 = \frac{\left( \frac{7\log_3(4)}{5} \right)^2 + 10}{3}
\]

7. **Take the square root to solve for \( x \):**
\[
x = \pm \sqrt{\frac{\left( \frac{7\log_3(4)}{5} \right)^2 + 10}{3}}
\]

At this point, you can plug in the values and calculate \( \log_3(4) \) to find the numerical values of \( x \).

However, without a calculator, you cannot simplify further in terms of numerical values. Thus, the solution in radical form is:
\[
x = \pm \sqrt{\frac{\left( \frac{7\log_3(4)}{5} \right)^2 + 10}{3}}
\]

If you need precise numerical values, you may need to compute \( \log_3(4) \) first, which can be estimated based on logarithmic properties.