Solve for x x^(2/3)=4x^(1/3)

Solution:

Step 1

Raise both sides of the equation to the power of 3 to remove the fractional exponents: $(x^{\frac{2}{3}})^3 = (4x^{\frac{1}{3}})^3$.

Step 2

Apply the exponentiation rule to simplify the left side: $x^{\frac{2}{3} \cdot 3} = (4x^{\frac{1}{3}})^3$.

Step 2.1

Utilize the power of a power property: $x^{2} = (4x^{\frac{1}{3}})^3$.

Step 2.2

Simplify the right side by raising both terms inside the parentheses to the power of 3.

Step 3

Expand the right side of the equation: $x^{2} = 4^3 (x^{\frac{1}{3}})^3$.

Step 3.1

Calculate the cube of 4: $x^{2} = 64(x^{\frac{1}{3}})^3$.

Step 3.2

Simplify the exponent on x: $x^{2} = 64x^{\frac{1}{3} \cdot 3}$.

Step 3.3

Finalize the simplification: $x^{2} = 64x$.

Step 4

Subtract $64x$ from both sides to set the equation to zero: $x^{2} - 64x = 0$.

Step 5

Factor out the common x term: $x(x - 64) = 0$.

Step 6

Apply the zero-product property: if $ab = 0$, then either $a = 0$ or $b = 0$.

Step 7

Solve for x when the first factor equals zero: $x = 0$.

Step 8

Solve for x when the second factor equals zero: $x - 64 = 0$, then $x = 64$.

Step 9

Combine the solutions from steps 7 and 8: The solutions are $x = 0$ and $x = 64$.

Knowledge Notes:

1. Fractional Exponents: The expression $x^{\frac{a}{b}}$ is equivalent to the b-th root of $x$ raised to the power of a, i.e., $(\sqrt[b]{x})^a$.

2. Power Rule: When raising a power to a power, you multiply the exponents, as in $(a^m)^n = a^{m \cdot n}$.

3. Zero-Product Property: If the product of two factors is zero, at least one of the factors must be zero. This property is used to solve quadratic equations by factoring.

4. Factoring: The process of breaking down an expression into a product of simpler expressions. In the case of $x^2 - 64x$, factoring out x gives $x(x - 64)$.

5. Solving Quadratic Equations: A quadratic equation can be solved by factoring, completing the square, using the quadratic formula, or graphing. In this case, factoring is the chosen method.

6. LCD (Least Common Denominator): The smallest common multiple of the denominators of two or more fractions. It is used to eliminate fractions by multiplying each term by the LCD.

7. Cube of a Number: The cube of a number is its third power, the result of multiplying it by itself three times. For example, $4^3 = 4 \cdot 4 \cdot 4 = 64$.

8. Simplification: The process of reducing an expression to its simplest form by performing all possible operations and combining like terms.