Question

If $2^x × 8^{(1/4) }= 2^(1/4)$ then find the value of x

• A. -1/2
• B. 1/2
• C. -1/4
• D. 1/4

Answer 1

Answer:

A. -1/2

Fundamental Laws of Exponents to Remember:

1. Product Rule: When multiplying powers with the same base, add the exponents.$a^m \times a^n = a^{(m+n)}$

2. Power of a Power Rule: When raising a power to another power, multiply the exponents.$(a^m)^n = a^{(m \times n)}$

3. Equating Exponents: If two powers with the same non-zero, non-one base are equal, then their exponents must also be equal.If $a^m = a^n$ (where $a \neq 0, 1, -1$), then $m = n$.

Step-by-Step Problem Solving Strategy:

• Step 1: Convert all bases to the lowest common base. Here, $8$ is converted to $2^3$.

• Step 2: Apply the 'Power of a Power' rule: For the term $8^{(1/4)}$, substitute $8$ with $2^3$. This yields $(2^3)^{(1/4)}$.

• Step 3: Simplify the exponent: Using the rule $(a^m)^n = a^{(m \times n)}$, $(2^3)^{(1/4)}$ becomes $2^{(3 \times 1/4)} = 2^{(3/4)}$.

• Step 4: Rewrite the original equation with the unified base: The equation $2^x \times 8^{(1/4)} = 2^{(1/4)}$ transforms into $2^x \times 2^{(3/4)} = 2^{(1/4)}$.

• Step 5: Apply the 'Product Rule' for exponents: Combine the terms on the left side of the equation. Using $a^m \times a^n = a^{(m+n)}$, $2^x \times 2^{(3/4)}$ becomes $2^{(x + 3/4)}$.

• Step 6: Equate the exponents: Now the equation is $2^{(x + 3/4)} = 2^{(1/4)}$. Since the bases are identical, we can set the exponents equal to each other: $x + 3/4 = 1/4$.

• Step 7: Solve the linear equation for x: Subtract $3/4$ from both sides: $x = 1/4 - 3/4 = -2/4 = -1/2$.