Radicals and rationals exponents quick check using the properties of exponents to generate an equivalent expression which of the following would you be the rational exponent on the expression equivalent to 3√ 216 3 - 1/3 1/3 -

Answer 1

To express the cube root of \(216\) using rational exponents, we can use the property of exponents that states:

\[ \sqrt[n]{a} = a^{1/n} \]

Thus, the cube root of \(216\) can be expressed as:

\[ 3\sqrt{216} = 216^{1/3} \]

Now, we can evaluate \(216\). We know that \(216 = 6^3\), so:

\[ 216^{1/3} = (6^3)^{1/3} \]

Applying the power of a power property \( (a^m)^n = a^{m \cdot n} \):

\[ (6^3)^{1/3} = 6^{3 \cdot (1/3)} = 6^1 = 6 \]

So, \(3\sqrt{216} = 6\).

If we are considering the question in relation to rational exponents, since \(3\sqrt{216}\) corresponds to \(216^{1/3}\).

In summary:

• The expression \(3\sqrt{216}\) can be rewritten as \(216^{1/3}\).
• Thus, the rational exponent corresponding to the expression \(3\sqrt{216}\) is \(1/3\).