Question

Rewrite the following expressions 27^1/2 * 27^1/4

\root(4)(27)
\root(3)(27)
27^3/4
27^1/8

Answer 1

To rewrite the expression \( 27^{1/2} \times 27^{1/4} \), we can use the property of exponents that states \( a^m \times a^n = a^{m+n} \).

So:

\[
27^{1/2} \times 27^{1/4} = 27^{1/2 + 1/4}
\]

To add the exponents, we need a common denominator. The common denominator for 2 and 4 is 4:

\[
1/2 = 2/4 \quad \text{and} \quad 1/4 = 1/4
\]

Now add them:

\[
1/2 + 1/4 = 2/4 + 1/4 = 3/4
\]

Thus, we have:

\[
27^{1/2} \times 27^{1/4} = 27^{3/4}
\]

The rewritten expression in list form is:

1. \( \sqrt[4]{27} \) (This is not equivalent to our expression)
2. \( \sqrt[3]{27} \) (This is equivalent to \( 27^{1} = 27 \) )
3. \( 27^{3/4} \) (This matches our computed result)
4. \( 27^{1/8} \) (This is not equivalent to our expression)

Therefore, the only correct rewriting of \( 27^{1/2} \times 27^{1/4} \) is:

**3. \( 27^{3/4} \)**.