| Key Takeaways: Solving x*x*x = 2 |
| ✓ x*x*x means x multiplied by itself three times, so the equation becomes x³ = 2. |
| ✓ The real solution is x = ∛2, the cube root of 2, which is approximately 1.2599210499. |
| ✓ Because 2 is not a perfect cube, ∛2 is irrational: its decimal expansion neither terminates nor repeats. |
| ✓ Over the real numbers, x³ = 2 has one solution. Over the complex numbers, it has two additional complex-conjugate roots. |
| ✓ You can approximate the cube root of 2 by estimation, logarithms, Newton’s method, or a calculator. |
What Does x*x*x = 2 Mean?
For x*x*x = 2, rewrite the repeated multiplication as x3 = 2, so the real solution is x = ∛2 ≈ 1.2599210499.
The asterisk (*) denotes multiplication. Therefore, x*x*x means that x is multiplied by itself three times:
x × x × x = 2 → x3 = 2
This is a cubic equation because the highest exponent on the variable is 3. To isolate x, take the cube root of both sides:
x = ∛2 = 21/3
The cube root of 2 is not a whole number or a terminating decimal, so its decimal form is approximate. For calculations, use x ≈ 1.2599210498948732. In other words, multiplying 1.2599210498948732 by itself three times gives a result approximately equal to 2.
Understanding What x*x*x = 2 Means
The expression x*x*x = 2 means that x is multiplied by itself three times, and the product is 2. In standard mathematical notation, the same equation is written as:
x × x × x = 2 → x3 = 2
The Asterisk Represents Multiplication
The asterisk (*) is another way to write the multiplication symbol, especially in calculators, spreadsheets, and programming. Therefore:
x*x*x = x × x × x = x3
Why x3 Is Called a Cubic Expression
The exponent 3 means that x is raised to the third power, or x cubed. Because the highest power of the variable is 3, x3 = 2 is a simple cubic equation. Solving it means finding the number whose cube equals 2.
What Does the Asterisk (*) Mean?
The asterisk (*) is a multiplication symbol commonly used in calculators, spreadsheets, programming, and plain-text math. Therefore, x*x*x means:
x × x × x = x³
In words, x is multiplied by itself three times. The exponent 3 in x³ shows how many factors of x are being multiplied. This is called x cubed.
Do not confuse x*x*x with 3x. The first expression means x × x × x, while 3x means 3 × x.
What Is a Cubic Equation?
A cubic equation is a polynomial equation whose highest exponent is 3. Its standard form is ax³ + bx² + cx + d = 0, where a cannot equal zero. The equation x³ = 2 can be rearranged as:
x³ − 2 = 0
Here, a = 1, b = 0, c = 0, and d = −2. It is a particularly simple cubic equation because it contains only the cubed variable and a constant.
How to Solve x*x*x = 2 Step by Step
To solve x*x*x = 2, rewrite it as x3 = 2 and take the cube root of both sides. Therefore, x = ∛2 ≈ 1.259921.
Step 1: Rewrite repeated multiplication as a power
The expression x*x*x means that x is multiplied by itself three times. In standard mathematical notation:
x*x*x = x3
So the original equation becomes:
x3 = 2
Step 2: Take the cube root of both sides
The inverse operation of cubing a number is taking its cube root. Apply the cube root to both sides:
∛(x3) = ∛2
Because the cube root reverses the third power, the left side simplifies to x:
x = ∛2 = 21/3
Step 3: Find the decimal approximation
The cube root of 2 is irrational, so it cannot be written as a terminating or repeating decimal. Its useful decimal approximation is:
x ≈ 1.2599210499
Step 4: Check the result
Substituting the approximation back into the equation gives:
1.2599210499 × 1.2599210499 × 1.2599210499 ≈ 2
Thus, the real solution to x*x*x is equal to 2 is x = ∛2 ≈ 1.259921.
Step 1: Express the Multiplication as a Power
The notation x*x*x means that x is multiplied by itself three times. In exponent form, this is written as x3, or “x cubed.” Therefore, the original equation becomes:
x3 = 2
Step 2: Take the Cube Root of Both Sides
The inverse operation of cubing a number is taking its cube root. Apply the cube root to both sides of x³ = 2 to isolate x:
∛(x³) = ∛2
Therefore, the real solution is:
x = ∛2 = 21/3
Step 3: Approximate the Cube Root of 2
The exact solution is x = ∛2. Because the cube root of 2 is irrational, its decimal form continues indefinitely, so use a rounded approximation for calculations:
x = ∛2 ≈ 1.2599210498948732
To four decimal places, this is x ≈ 1.2599. Cubing the rounded value gives a result very close to 2.
Step 4: Verify the Cube Root
Substitute the approximation back into the original equation:
1.2599210499 × 1.2599210499 × 1.2599210499 ≈ 2
Because the rounded value produces approximately 2, it verifies that x = ∛2 ≈ 1.2599210499 solves x³ = 2. The exact solution remains ∛2, since its decimal expansion is irrational.
Why Is ∛2 Not a Whole Number?
The equation x³ = 2 does not produce a whole-number answer because 2 is not a perfect cube. A perfect cube is the result of multiplying an integer by itself three times, such as 1³ = 1, 2³ = 8, or 3³ = 27.
Because 2 lies between 1 and 8, its cube root lies between 1 and 2:
1 < ∛2 < 2
The exact solution is x = ∛2, and its decimal approximation is 1.2599210498948732. This value is irrational, meaning it cannot be written as a fraction of two integers. Its decimal expansion continues indefinitely without repeating, so no finite decimal or simple fraction represents ∛2 exactly.
Perfect Cubes: Reference Values for x³
A perfect cube is an integer that can be written as x³, where x is also an integer. The equation x³ = 2 has no whole-number solution because 2 does not appear among the perfect cube values. Since 1³ = 1 and 2³ = 8, the value of ∛2 lies between 1 and 2.
| Value of x | x³ (x cubed) |
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
Because 2 falls between the consecutive cubes 1 and 8, its cube root is not an integer. The real solution is therefore the irrational number ∛2 ≈ 1.259921, rather than a whole number such as 1 or 2.
Methods to Find the Cube Root of 2
To solve x*x*x = 2, find the number whose cube equals 2. The exact answer is x = ∛2 = 21/3, and its decimal approximation is 1.2599210499. You can find this cube root by estimation, logarithms, Newton’s method, or a calculator.
Estimate Between Perfect Cubes
Because 1³ = 1 and 2³ = 8, ∛2 must lie between 1 and 2. Test increasingly precise values:
- 1.25³ = 1.953125, which is less than 2.
- 1.26³ = 2.000376, which is slightly greater than 2.
Therefore, the answer is between 1.25 and 1.26. Narrowing the interval gives ∛2 ≈ 1.259921.
Use Logarithms
Starting with x³ = 2, take the natural logarithm of both sides:
3 ln(x) = ln(2)
Divide by 3 and exponentiate:
x = eln(2)/3 = 21/3 ≈ 1.259921
This method is useful when a calculator or mathematical software provides logarithm and exponential functions.
Apply Newton’s Method
For a hand-calculated numerical approximation, define f(x) = x³ − 2. Newton’s method uses the iteration:
xn+1 = xn − (xn³ − 2)/(3xn²)
Beginning with a reasonable estimate such as x0 = 1.25, repeated substitutions quickly approach:
x ≈ 1.2599210499
Use a Calculator or Computer
On a scientific calculator, enter the cube-root function or calculate 21/3. In Python, the expression 2 ** (1/3) returns approximately 1.2599210498948732.
Method 1: Estimate the Cube Root by Narrowing the Range
Because \(1^3 = 1\) and \(2^3 = 8\), the value of \(\sqrt[3]{2}\) must be between 1 and 2. Test decimal values and compare their cubes with 2:
- 1.25³ = 1.953125, which is less than 2.
- 1.26³ = 2.000376, which is slightly greater than 2.
Therefore, \(\sqrt[3]{2}\) lies between 1.25 and 1.26. Refining the estimate gives:
x = \sqrt[3]{2} \approx 1.259921
This trial-and-error approach works by repeatedly narrowing the interval until the desired level of accuracy is reached.
Method 2: Solve \(x^3 = 2\) with Logarithms
For the real solution, \(x\) is positive, so you can take the natural logarithm of both sides:
\(\ln(x^3) = \ln(2)\)
Using the logarithm power rule, \(\ln(x^3) = 3\ln(x)\). Therefore:
3\ln(x) = \ln(2)
Divide by 3 and then exponentiate both sides:
\(\ln(x) = \frac{\ln(2)}{3}\)
\(x = e^{\ln(2)/3} = 2^{1/3} = \sqrt[3]{2}\)
Since \(\ln(2) \approx 0.693147\), this gives:
\(x \approx e^{0.231049} \approx 1.259921\)
Logarithms provide an exact expression and a reliable way to calculate the decimal approximation of the cube root of 2.
Method 3: Newton’s Method for Approximating ∛2
Newton’s method, also known as the Newton–Raphson method, provides a fast way to approximate the real solution of x³ = 2. Define the function:
f(x) = x³ − 2
Because f′(x) = 3x², Newton’s formula becomes:
xn+1 = xn − (xn³ − 2)/3xn²
Starting with the estimate x0 = 1.5, each iteration produces a more accurate value:
- x1 ≈ 1.296296
- x2 ≈ 1.260933
- x3 ≈ 1.259921
Thus, Newton’s method quickly converges to the real cube root:
x = ∛2 ≈ 1.2599210499
This approach is especially useful when a calculator or computer needs to calculate a cube root numerically rather than using a dedicated cube-root function.
Method 4: Use a Calculator or Computer Program
A scientific calculator can find the cube root of 2 directly. Enter ∛2, or use the equivalent exponent form, 21/3, to obtain:
x = ∛2 ≈ 1.2599210498948732
You can also calculate the value with code. In Python, use:
x = 2 ** (1 / 3)
print(x)This returns approximately 1.2599210498948732. The result is a decimal approximation because the cube root of 2 is irrational.
Real and Complex Solutions of x3 = 2
The equation x3 = 2 has one real solution and two nonreal complex solutions. The real answer is x = ∛2 ≈ 1.259921, while the other two roots are complex conjugates.
To find all three roots, write 2 as (∛2)3 and multiply the cube root by the three cube roots of 1: 1, −1/2 + i√3/2, and −1/2 − i√3/2. Here, i2 = −1.
| Root type | Solution |
| Real root | x = ∛2 ≈ 1.259921 |
| Complex root 1 | x = ∛2 × (−1/2 + i√3/2) |
| Complex root 2 | x = ∛2 × (−1/2 − i√3/2) |
In elementary algebra and most practical calculations, “x*x*x is equal to 2” usually asks for the real value, so the expected answer is x = ∛2 ≈ 1.259921. The complex roots become important when studying polynomial roots, complex numbers, or advanced engineering and physics.
Why the Cube Root of 2 Matters in Mathematical History
The number ∛2 is closely associated with the ancient Greek problem known as doubling the cube, or the Delian problem. The goal was to construct a cube with twice the volume of a given cube using only a straightedge and compass.
If the original cube has side length 1, a cube with twice its volume must have a side length ∛2, because:
(∛2)³ = 2
Although ∛2 can be defined precisely, it cannot be constructed with only the classical straightedge-and-compass tools. The impossibility was established in the 19th century using the algebraic properties of the number. This result helped demonstrate the limits of geometric construction methods.
Doubling the cube was one of three famous unsolved construction problems from antiquity, along with trisecting an arbitrary angle and squaring the circle. These challenges encouraged mathematicians to investigate irrational numbers, equations, and the relationship between geometry and algebra.
Where Cube Roots and Cubic Equations Are Used
The equation x3 = 2 models situations in which one quantity changes with the cube of another. Its real-world solution, x = ∛2 ≈ 1.259921, is especially useful when a three-dimensional measurement must be adjusted while preserving a proportional relationship.
Engineering and Construction
For a cube, volume is calculated as V = s3, where s is the side length. If a cubic container must hold twice as much volume, its new side length is:
snew = ∛2 × soriginal ≈ 1.2599 × soriginal
That means doubling the volume does not require doubling each dimension. Increasing every side by about 25.99% produces twice the volume, assuming the container remains geometrically similar.
Physics and Scientific Measurement
Cube roots help scientists recover a length scale from a volume or another quantity proportional to volume. For example, the characteristic radius of a spherical object can be found from its volume using a cube-root relationship. Similar reasoning appears in dimensional analysis, where a measured three-dimensional quantity is converted into a representative length.
Chemistry and Crystallography
In chemistry and materials science, cube-root calculations can connect the volume of a unit cell with its approximate linear dimensions. The specific equation depends on the crystal structure, so not every crystallography problem is simply x3 = 2; however, the same principle applies whenever volume scales with the third power of length.
Computer Graphics and 3D Modeling
Three-dimensional software uses scaling factors to resize objects. When a model must retain its proportions while its volume changes by a specified factor, the required uniform scale is the cube root of that factor. For a volume increase by 2, the uniform scale factor is ∛2, not 2.
Finance and Other Cubic Models
Some financial models use cubic equations when a variable appears to the third power, but ordinary compound-interest formulas are usually exponential rather than cubic. A cubic equation may arise after rearranging a specialized model or when estimating a rate in a three-period relationship. The correct method depends on the equation being solved, so x3 = 2 should be treated as a simple example of a broader cubic model.
Common Errors When Solving x*x*x = 2
The most common mistakes involve interpreting multiplication, choosing the wrong root, or overlooking the difference between real and complex solutions. Use these checks to keep the solution accurate.
- Confusing x*x*x with 3x: Repeating multiplication is not the same as multiplying by 3. The expression x*x*x means x3. For example, when x = 4, x3 = 64, whereas 3x = 12.
- Using a square root instead of a cube root: To solve x3 = 2, apply the cube root to both sides: x = ∛2 ≈ 1.259921. The value √2 ≈ 1.414214 solves x2 = 2, not x3 = 2.
- Rounding too early: The real solution is irrational, so its decimal expansion does not terminate or repeat. Keep the exact form ∛2 during the calculation and round to a stated precision, such as 1.260 or 1.259921.
- Assuming the cube root of 2 is a fraction: ∛2 cannot be written as a ratio of integers. It is irrational because 2 is not a perfect cube; the neighboring integer cubes are 13 = 1 and 23 = 8.
- Listing only one root in a complex-number context: Over the real numbers, x = ∛2 is the only solution. Over the complex numbers, the cubic equation also has two nonreal conjugate roots, ∛2(−1/2 + i√3/2) and ∛2(−1/2 − i√3/2).
Related Equations and Their Solutions
Comparing x*x*x = 2 with nearby power equations helps show how square roots, cube roots, and fourth roots differ. The table lists the real solutions most commonly used in algebra.
| Equation | Real solution |
| x*x = 2 (x² = 2) | x = ±√2 ≈ ±1.4142 |
| x*x*x = 3 (x³ = 3) | x = ∛3 ≈ 1.4422 |
| x*x*x = 8 (x³ = 8) | x = 2, because 8 is a perfect cube |
| x*x*x = 27 (x³ = 27) | x = 3, because 27 is a perfect cube |
| x*x*x*x = 2 (x⁴ = 2) | x = ±⁴√2 ≈ ±1.1892 |
| x*x*x = −1 (x³ = −1) | x = −1 |
The key pattern is that an equation of the form xn = a is solved by taking the nth root of both sides. For odd powers such as x³, every real number has one real root. For even powers such as x² and x⁴, a positive result usually produces two real solutions: one positive and one negative.
Cube Root vs. Square Root: What Is the Difference?
Square roots and cube roots use different powers: √x asks which number multiplied by itself equals x, while ∛x asks which number multiplied by itself three times equals x. Therefore, x × x × x = 2 requires a cube root, not a square root.
| Feature | Square Root | Cube Root |
| Notation | √x or x1/2 | ∛x or x1/3 |
| Meaning | A number multiplied by itself equals x | A number multiplied by itself three times equals x |
| Example | √4 = 2 because 2 × 2 = 4 | ∛8 = 2 because 2 × 2 × 2 = 8 |
| Negative real inputs | √x is not a real number when x < 0 | ∛x is a real number when x is negative |
| Value for 2 | √2 ≈ 1.4142 | ∛2 ≈ 1.2599 |
The difference also affects the number of real solutions. For example, x2 = 4 has two real solutions, x = 2 and x = −2, whereas x3 = 8 has one real solution, x = 2. In the equation x3 = 2, the required value is x = ∛2 ≈ 1.259921.
Frequently Asked Questions About x*x*x = 2
What is the value of x when x*x*x equals 2?
The real solution is x = ∛2, the cube root of 2. In decimal form, x ≈ 1.2599210499.
How do you write x*x*x in mathematical notation?
x*x*x means that x is multiplied by itself three times. It is written as x3, or “x cubed.” Therefore, x*x*x = 2 becomes x3 = 2.
How do you solve x3 = 2?
Take the cube root of both sides:
x = ∛2 = 21/3 ≈ 1.2599210499
Is the cube root of 2 rational or irrational?
∛2 is irrational. It cannot be expressed as a fraction of two integers, so its decimal expansion continues indefinitely without repeating.
Why is ∛2 between 1 and 2?
Because 13 = 1 and 23 = 8. Since 2 lies between 1 and 8, its cube root must lie between 1 and 2.
Is x*x*x the same as 3x?
No. x*x*x means x multiplied by itself three times, or x3. By contrast, 3x means 3 multiplied by x. For example, when x = 2, x*x*x = 8, while 3x = 6.
How many solutions does x3 = 2 have?
Over the real numbers, there is one solution: x = ∛2. Over the complex numbers, the cubic has three roots in total: one real root and two complex conjugate roots.
Can you find the cube root of 2 without a calculator?
Yes. Since ∛2 is between 1 and 2, estimate by testing nearby values. For example, 1.253 = 1.953125 and 1.263 = 2.000376, so ∛2 is approximately 1.2599.
What is the value of x if x*x*x is equal to 2?
Rewrite x*x*x = 2 as x3 = 2. Taking the cube root of both sides gives x = ∛2 = 21/3, so the real solution is approximately x = 1.2599210498948732.
Is ∛2 Rational or Irrational?
∛2 is irrational. It cannot be expressed as a fraction of two integers. If ∛2 were rational, then writing it in lowest terms as a/b would imply a3 = 2b3, which is impossible because the prime factorization of the left side has exponents divisible by 3, while the right side contains one additional factor of 2. Therefore, ∛2 has a nonterminating, nonrepeating decimal expansion: approximately 1.259921.
What Is x*x*x in Mathematical Notation?
x*x*x means multiplying x by itself three times. In standard mathematical notation, it is written as x3, read as “x cubed” or “x raised to the third power.”
How many solutions does x3 = 2 have?
The equation x3 = 2 has three solutions over the complex numbers: one real solution and two nonreal complex conjugates. The real solution is x = ∛2 ≈ 1.259921. In general, a cubic polynomial has three complex roots when multiplicities are counted.
Can I solve x*x*x = 2 without a calculator?
Yes. Rewrite the equation as x3 = 2, so the exact solution is x = ∛2. To estimate it by hand, note that 13 = 1 and 23 = 8, which places x between 1 and 2. Testing narrower values gives 1.253 = 1.953125 and 1.263 = 2.000376, so ∛2 ≈ 1.2599. Newton’s method can produce additional decimal places, but no finite decimal equals ∛2 because it is irrational.
What Is the Difference Between x × x × x and 3x?
x × x × x means that x is multiplied by itself three times, so it is written as x3 or “x cubed.” By contrast, 3x means that x is multiplied by 3. They are different expressions: if x = 2, then x3 = 8, while 3x = 6.
Where Is x3 = 2 Used in Real Life?
The equation x3 = 2 is useful whenever a quantity changes with the cube of a length or scale. Its real solution, x = ∛2 ≈ 1.2599, means that increasing a cube’s side length by about 26% doubles its volume.
- Engineering and design: Find the side length of a cubic container with twice the volume of a reference container.
- Physics: Model relationships involving three-dimensional volume, scale, or other cubic quantities.
- Computer graphics: Apply three-dimensional scaling when an object’s volume must increase by a specific factor.
- Applied mathematics: Use cube roots to solve cubic models that arise in optimization, geometry, and numerical analysis.
Conclusion
To solve x*x*x = 2, rewrite it as x³ = 2 and take the cube root of both sides:
x = ∛2 = 21/3 ≈ 1.2599210499
Therefore, the real value of x is approximately 1.2599. Because 2 is not a perfect cube, its cube root is irrational, so the decimal continues without terminating or repeating.
The key distinction is that x*x*x means multiplying x by itself three times—not multiplying x by 3. Substituting the result confirms the solution: (∛2)³ = 2. Over the complex numbers, the cubic equation also has two additional complex roots, but ∛2 is the only real solution.
