Solve each equation. 3x^3/4= x^1/2

Verified step by step guidance

Rewrite the equation $3x^{3/4} = x^{1/2}$ and recognize that both sides have expressions with exponents involving $x$.

To solve for $x$, first bring all terms to one side to set the equation equal to zero: $3x^{3/4} - x^{1/2} = 0$.

Factor out the common term with the smallest exponent, which is $x^{1/2}$, so the equation becomes $x^{1/2}(3x^{1/4} - 1) = 0$.

Set each factor equal to zero separately: $x^{1/2} = 0$ and $3x^{1/4} - 1 = 0$.

Solve each equation for $x$: For $x^{1/2} = 0$, raise both sides to the power of 2; for $3x^{1/4} - 1 = 0$, isolate $x^{1/4}$ and then raise both sides to the power of 4.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponents and Rational Exponents

Exponents indicate repeated multiplication, and rational exponents represent roots; for example, x^(1/2) means the square root of x. Understanding how to manipulate and simplify expressions with fractional exponents is essential for solving equations involving roots and powers.

Solving Equations Involving Exponents

To solve equations with exponents, isolate the variable term and use properties of exponents to rewrite expressions with a common base or exponent. This often involves raising both sides to a power that eliminates the fractional exponent or rewriting terms to compare powers directly.

Domain Restrictions for Variables with Rational Exponents

When variables have fractional exponents with even denominators, the base must be non-negative to keep the expression real. Recognizing domain restrictions ensures solutions are valid within the real number system and prevents extraneous or undefined solutions.

Recommended video:

04:06

Rational Exponents

Related Practice

Textbook Question

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) 4x² = -6x + 3

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (9x-11)(2x+7)/(3x-8)³>0

Textbook Question

Solve each equation. x^2/3 = 2x^1/3

views

Textbook Question

Solve each equation. 2x⁴-7x²+5=0

views

Textbook Question

Solve each inequality. Give the solution set using interval notation.

$7 x - 2 (x - 3) \leq 5 (2 - x)$