Question

The value of \( 3^{1/4} \times 27^{0.25} \) is:

• A. 6
• B. 27
• C. 3
• D. 9

Hint:

An alternative quick rule when exponents are identical is:
\[ a^x \times b^x = (a \times b)^x \] Here:
\[ 3^{1/4} \times 27^{1/4} = (3 \times 27)^{1/4} = (81)^{1/4} = (3^4)^{1/4} = 3^1 = 3 \] This approach avoids working with fractions and gets you the answer in seconds.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires the application of the laws of exponents.
When multiplying expressions with the same base, their exponents are added.
Additionally, we can convert terms with different bases to a common base if they are powers of the same number.
Key Formula or Approach:
The relevant algebraic identities of exponents are:
1. \( (a^m)^n = a^{m \cdot n} \)
2. \( a^m \times a^n = a^{m + n} \)
3. Conversion of decimals to fractions: \( 0.25 = \frac{25}{100} = \frac{1}{4} \)

Step 2: Detailed Explanation:
Let us simplify the given expression step-by-step:
Given expression:
\[ E = 3^{1/4} \times 27^{0.25} \] 1. Express the decimal exponent as a fraction:
\[ 0.25 = \frac{1}{4} \] Substitute this back into the expression:
\[ E = 3^{1/4} \times 27^{1/4} \] 2. Express the base 27 as a power of 3:
We know that:
\[ 27 = 3 \times 3 \times 3 = 3^3 \] Substitute this into the expression:
\[ 27^{1/4} = (3^3)^{1/4} = 3^{3/4} \] 3. Rewrite the entire product with the common base of 3:
\[ E = 3^{1/4} \times 3^{3/4} \] 4. Apply the law of exponents by adding the powers:
\[ E = 3^{(1/4 + 3/4)} \] Combine the fractional exponents:
\[ \frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1 \] Therefore:
\[ E = 3^1 = 3 \] The simplified value of the expression is 3.

Step 3: Final Answer:
The correct option is (C).