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Solve the equation using inverse operations. Check your solutions. In your final answer, include all of your work.

2+x^3= 18

7 answers

To solve the equation \(2 + x^3 = 18\) using inverse operations, follow these steps:

Step 1: Isolate the term with the variable (x^3)

Subtract 2 from both sides of the equation:

\[ 2 + x^3 - 2 = 18 - 2 \]

This simplifies to:

\[ x^3 = 16 \]

Step 2: Take the cube root of both sides

To solve for \(x\), take the cube root of both sides:

\[ x = \sqrt[3]{16} \]

Step 3: Simplify the cube root

To simplify \(\sqrt[3]{16}\):

We know that \(16 = 2^4\). Thus,

\[ x = \sqrt[3]{2^4} = 2^{4/3} = 2^{1 \frac{1}{3}} = 2 \cdot 2^{1/3} \] So if necessary, we can express \(x\) in decimal form, but we leave it as \(2^{4/3}\) for exactness.

Step 4: Check the solution

To verify, substitute \(x = \sqrt[3]{16}\) back into the original equation:

\[ 2 + (\sqrt[3]{16})^3 = 2 + 16 = 18 \]

Since both sides of the equation are equal, the solution is verified.

Final Answer

The solution to the equation \(2 + x^3 = 18\) is:

\[ x = \sqrt[3]{16} \]

In approximate decimal form, \(x \approx 2.52\) (but the exact answer is preferred). Thus, the final answer can be written as:

\[ x = 2^{4/3} \quad \text{or} \quad x = \sqrt[3]{16} \]

can you do it without all the symbols and slashes

Of course! Let's solve the equation step by step without using symbols and slashes.

Step 1: Start with the original equation

The equation we need to solve is:

Two plus \(x\) cubed equals eighteen.

Step 2: Isolate the term with \(x\)

To do this, we will subtract two from both sides of the equation:

When we subtract two from two, we get zero.
When we subtract two from eighteen, we get sixteen.

Now we have:

\(x\) cubed equals sixteen.

Step 3: Solve for \(x\)

To find \(x\), we need to take the cube root of sixteen.

The cube root of sixteen can be thought of as the number that, when multiplied by itself three times, gives us sixteen.

We know that sixteen is equal to two multiplied by two multiplied by two multiplied by two, or two to the fourth power.

So, the cube root of sixteen can also be expressed as two raised to the four-thirds power or rewritten in a simpler form.

Step 4: Check the solution

Let's verify our solution by substituting back into the original equation.

If we take the cube root of sixteen and then cube it, we should get sixteen back.

When we add two to that, we should get eighteen.

Verification:

  • Cube the number we found (the cube root of sixteen) to see if it equals sixteen.
  • Then we add two to that result.

Since both sides balance (two plus sixteen equals eighteen), our solution is confirmed.

Final Answer

The solution to the equation is:

\(x\) equals the cube root of sixteen, or written as a power, \(x\) equals two raised to the four-thirds power.

can you use more numbers and make it a little less advanced

Sure! Let’s break it down in a very simple way using numbers, without complicated language.

Step 1: Start with the equation

We have:

2 + x³ = 18

Step 2: Get x³ by itself

To do this, we need to get rid of the 2 on the left side. We can do that by subtracting 2 from both sides of the equation.

So, we subtract 2 from 2 and 2 from 18:

2 - 2 + x³ = 18 - 2

That simplifies to:

x³ = 16

Step 3: Find x

Now we need to find x. To do this, we need to figure out what number when multiplied by itself three times gives us 16.

This is called the cube root. The number we are looking for is:

x = cube root of 16

If we want to make it easier to find, we can think of it this way:

We know that 2 × 2 × 2 = 8 (which is 2 cubed) and 3 × 3 × 3 = 27 (which is 3 cubed). But 4 × 4 × 4 = 64 (which is 4 cubed).

So, the number we are looking for lies somewhere between 2 and 3.

To be more precise, we can express it as:

x ≈ 2.52

Step 4: Verify the solution

Let's check if our answer is correct.

  1. First, we will cube our answer \( (cube root of 16) \):

    • x ≈ 2.52 (when we cube this, we should get 16).

  2. Then, we will add 2:

    • 2 + 16 = 18.

Since both sides of the original equation match (2 + 16 = 18), our solution is correct.

Final Answer

So, in simple terms, the solution is:

x = cube root of 16, or about 2.52.

can you use less words

Sure! Here’s the solution with fewer words:

Step 1: Start with the equation

2 + x³ = 18

Step 2: Isolate x³

Subtract 2 from both sides:

x³ = 18 - 2
x³ = 16

Step 3: Find x

Take the cube root of 16:

x = cube root of 16
x ≈ 2.52

Step 4: Check

Check if it works:

2 + (2.52)³ ≈ 18
So, it checks out.

Final Answer

x ≈ 2.52 (or x = cube root of 16).