Use the properties of rational exponents to simplify $$ \frac{2^{2 / 3} \cdot 16^{2 / 3}}{4^{2 / 3}} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression $$ \frac{2^{2 / 3} \cdot 16^{2 / 3}}{4^{2 / 3}} $$, we'll use the properties of exponents:

1. **Express all bases as powers of 2:**
   - $$ 16 = 2^4 $$
   - $$ 4 = 2^2 $$

2. **Rewrite the expression using these equivalents:**
   $$
   \frac{2^{2/3} \cdot (2^4)^{2/3}}{(2^2)^{2/3}}
   $$

3. **Apply the power of a power property $$(a^m)^n = a^{mn}$$:**
   $$
   \frac{2^{2/3} \cdot 2^{(4 \cdot 2/3)}}{2^{(2 \cdot 2/3)}} = \frac{2^{2/3} \cdot 2^{8/3}}{2^{4/3}}
   $$

4. **Combine the exponents in the numerator using $$a^m \cdot a^n = a^{m+n}$$:**
   $$
   \frac{2^{(2/3 + 8/3)}}{2^{4/3}} = \frac{2^{10/3}}{2^{4/3}}
   $$

5. **Subtract the exponents in the division using $$a^m / a^n = a^{m-n}$$:**
   $$
   2^{10/3 - 4/3} = 2^{6/3} = 2^2
   $$

6. **Calculate the final value:**
   $$
   2^2 = 4
   $$

**Final Answer:**  
4
The simplified value of the expression is 4.

Use the properties of rational exponents to simplify

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Algebra Questions