get x + (24)^1/3 = (16)^1/3 + (36)^1/3 as fractions like 250/87 or something in the answer key

Let's start by simplifying each term. The equation is:

\[
x + \sqrt[3]{24} = \sqrt[3]{16} + \sqrt[3]{36}
\]

### Step 1: Express the cube roots in fractional form
We will approximate the cube roots of 24, 16, and 36 as fractions.

1. **Cube root of 24:**

\[
\sqrt[3]{24} \approx 2.884 \quad \text{(approximation, to be expressed as a fraction)}
\]

Using continued fraction approximation, this is roughly:

\[
\sqrt[3]{24} \approx \frac{361}{125}
\]

2. **Cube root of 16:**

\[
\sqrt[3]{16} = 2.5198421 \quad \text{(approximation)}
\]

This can be expressed as:

\[
\sqrt[3]{16} \approx \frac{161}{64}
\]

3. **Cube root of 36:**

\[
\sqrt[3]{36} \approx 3.301 \quad \text{(approximation)}
\]

This can be approximated as:

\[
\sqrt[3]{36} \approx \frac{3301}{1000}
\]

### Step 2: Plug the values into the equation
Now, rewrite the original equation using these fractions:

\[
x + \frac{361}{125} = \frac{161}{64} + \frac{3301}{1000}
\]

### Step 3: Combine the right-hand side
To combine the fractions on the right-hand side, find a common denominator:

The least common denominator of 64 and 1000 is 16000. Now rewrite the fractions:

\[
\frac{161}{64} = \frac{2515}{4000}, \quad \frac{3301}{1000} = \frac{13204}{4000}
\]

So:

\[
\frac{161}{64} + \frac{3301}{1000} = \frac{2515 + 13204}{4000} = \frac{15719}{4000}
\]

### Step 4: Solve for \( x \)
Now the equation is:

\[
x + \frac{361}{125} = \frac{15719}{4000}
\]

Subtract \(\frac{361}{125}\) from both sides:

To subtract, we need to express \(\frac{361}{125}\) with a denominator of 4000:

\[
\frac{361}{125} = \frac{11552}{4000}
\]

Now subtract:

\[
x = \frac{15719}{4000} - \frac{11552}{4000} = \frac{4167}{4000}
\]

### Step 5: Simplify the fraction
Simplify \(\frac{4167}{4000}\):

\[
x = \frac{4167}{4000}
\]

This fraction cannot be simplified further.

### Final Answer:
\[
x = \frac{4167}{4000}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How do you simplify cube roots as fractions?
2. What is the continued fraction approximation for cube roots?
3. How do you find a common denominator when adding fractions?
4. How do cube roots behave in equations involving addition or subtraction?
5. How do you solve equations with irrational numbers?

**Tip:** When approximating cube roots, continued fractions provide a more accurate representation of irrational numbers.

Learn how to solve the equation x + (24)^1/3 = (16)^1/3 + (36)^1/3 by approximating cube roots as fractions. This problem covers key algebraic concepts, including cube roots, fraction approximation, and equation solving. Suitable for students in Grades 10-12.

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