Question

kuta software - infinite algebra 2
simplifying rational exponents
simplify.

1. ((n^4)^{\frac{3}{2}})
2. ((25b^6)^{-1.5})
3. ((a^8)^{\frac{3}{2}})
4. ((81x^{12})^{1.25})

Explanation:

1. Simplify $(n^4)^{\frac{3}{2}}$

Step1: Apply power of a power rule

$(n^4)^{\frac{3}{2}} = n^{4 \times \frac{3}{2}}$

Step2: Calculate the exponent

$n^{4 \times \frac{3}{2}} = n^{6}$

3. Simplify $(25b^6)^{-1.5}$

Step1: Convert decimal to fraction

$(25b^6)^{-1.5} = (25b^6)^{-\frac{3}{2}}$

Step2: Apply negative exponent rule

$(25b^6)^{-\frac{3}{2}} = \frac{1}{(25b^6)^{\frac{3}{2}}}$

Step3: Split base and apply power rule

$\frac{1}{25^{\frac{3}{2}} \times (b^6)^{\frac{3}{2}}}$

Step4: Simplify each term

$25^{\frac{3}{2}} = (\sqrt{25})^3 = 5^3 = 125$, $(b^6)^{\frac{3}{2}} = b^{6 \times \frac{3}{2}} = b^9$

Step5: Combine results

$\frac{1}{125b^9}$

5. Simplify $(a^8)^{\frac{3}{2}}$

Step1: Apply power of a power rule

$(a^8)^{\frac{3}{2}} = a^{8 \times \frac{3}{2}}$

Step2: Calculate the exponent

$a^{8 \times \frac{3}{2}} = a^{12}$

7. Simplify $(81x^{12})^{1.25}$

Step1: Convert decimal to fraction

$(81x^{12})^{1.25} = (81x^{12})^{\frac{5}{4}}$

Step2: Split base and apply power rule

$81^{\frac{5}{4}} \times (x^{12})^{\frac{5}{4}}$

Step3: Simplify each term

$81^{\frac{5}{4}} = (\sqrt[4]{81})^5 = 3^5 = 243$, $(x^{12})^{\frac{5}{4}} = x^{12 \times \frac{5}{4}} = x^{15}$

Step4: Combine results

$243x^{15}$

Answer:

1. $n^6$
2. $\frac{1}{125b^9}$
3. $a^{12}$
4. $243x^{15}$