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.