Solve the Rational Equation for w ( fourth root of 7w-5)/2 = fourth root of w-2
The problem is asking you to find the value of the variable w that satisfies the given equation. The equation involves fourth roots, which makes it a type of radical equation. In this particular example, the fourth root of an expression involving w is equated to the fourth root of another expression involving w. Your task is to manipulate this equation algebraically to isolate w and find its value(s) that make the equation true. This typically involves eliminating the radical by raising both sides to the power that corresponds to the root, in this case, the fourth power, and then solving the resulting polynomial equation for w.
$\frac{\sqrt[4]{7 w - 5}}{2} = \sqrt[4]{w - 2}$
Answer
Solution:
Step 1: Isolate the fourth root term $\sqrt[4]{7w - 5}$.
1.1. Double each side of the equation to remove the denominator: $2 \cdot \frac{\sqrt[4]{7w - 5}}{2} = 2 \cdot \sqrt[4]{w - 2}$.
1.2. Simplify the equation by eliminating the common factor of 2: $\sqrt[4]{7w - 5} = 2 \cdot \sqrt[4]{w - 2}$.
Step 2: Eliminate the fourth root on the left side by raising both sides to the power of 4: $(\sqrt[4]{7w - 5})^4 = (2 \cdot \sqrt[4]{w - 2})^4$.
Step 3: Simplify both sides of the equation.
3.1. Convert the fourth root to a power: $(7w - 5)^{\frac{1}{4}}$.
3.2. Apply the exponent rule to simplify the left side: $(7w - 5)^{\frac{1}{4} \cdot 4} = 7w - 5$.
3.3. Simplify the right side of the equation:
3.3.1. Apply the power of 4 to the product: $2^4 \cdot (\sqrt[4]{w - 2})^4$.
3.3.2. Simplify the expression: $16 \cdot (w - 2)$.
3.3.3. Distribute the 16: $16w - 32$.
Step 4: Solve for $w$.
4.1. Move all $w$ terms to one side by subtracting $16w$ from both sides: $7w - 5 - 16w = -32$.
4.2. Move constants to the other side by adding 5 to both sides: $-9w = -32 + 5$.
4.3. Divide by the coefficient of $w$ to solve for $w$: $\frac{-9w}{-9} = \frac{-27}{-9}$, which simplifies to $w = 3$.
Knowledge Notes:
Rational Equations: Equations that involve rational expressions. Solving them typically involves finding a common denominator and making the denominators equal.
Radicals: Expressions that involve roots. The fourth root of a number $x$, denoted $\sqrt[4]{x}$, is a number that, when raised to the power of 4, equals $x$.
Exponent Rules: Important rules include $(a^m)^n = a^{m \cdot n}$ and $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. These rules are used to simplify expressions involving powers and roots.
Isolating the Variable: A common technique in algebra where you manipulate the equation to get the variable of interest by itself on one side of the equation.
Distributive Property: A property that states $a(b + c) = ab + ac$. It is used to multiply a single term and two or more terms inside a set of parentheses.
Simplifying Expressions: The process of reducing an expression to its simplest form, often by combining like terms and applying arithmetic operations.
