Solve (t-3)3/4=2 where t is a real number. Simplify your answer as much as possible. If there is more than one solution, separate them with commas

We are given the equation:

\[
(t - 3)^{3/4} = 2
\]

To solve for \( t \), follow these steps:

### Step 1: Eliminate the fractional exponent
Raise both sides of the equation to the power of \( \frac{4}{3} \) to eliminate the fractional exponent on the left-hand side:

\[
\left( (t - 3)^{3/4} \right)^{4/3} = 2^{4/3}
\]

Since \( \left( x^{a} \right)^{b} = x^{ab} \), the left-hand side simplifies to \( t - 3 \):

\[
t - 3 = 2^{4/3}
\]

### Step 2: Solve for \( t \)
Now add 3 to both sides:

\[
t = 2^{4/3} + 3
\]

### Step 3: Simplify \( 2^{4/3} \)
We can express \( 2^{4/3} \) as:

\[
2^{4/3} = \left( 2^4 \right)^{1/3} = 16^{1/3}
\]

Thus, the solution is:

\[
t = 16^{1/3} + 3
\]

This is the simplified form of the solution.

### Final Answer:
\[
t = 16^{1/3} + 3
\]

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How do you solve equations with fractional exponents?
2. What are the steps to simplifying expressions with radicals?
3. Can all radical equations be solved by raising both sides to a power?
4. How can you verify the solution of an equation with exponents?
5. How would the solution change if there were multiple terms involving \( t \)?

**Tip:** When dealing with fractional exponents, raising both sides to the reciprocal power is a quick way to eliminate the fraction!

Solve (t-3)^(3/4) = 2 where t is a real number. Simplify your answer as much as possible. If there is more than one solution, separate them with commas.

Learn how to solve the algebraic equation (t-3)^(3/4) = 2. This step-by-step solution involves exponentiation, radical simplification, and basic algebraic techniques. Perfect for high school students in Grades 10-12.

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