Question

kuta software - infinite alge
simplifying rational
simplify.

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

Explanation:

Step1: Apply power rule $(x^m)^n=x^{m \cdot n}$

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

Step2: Calculate exponent product

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

---

Step1: Convert decimal exponent to fraction

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

Step2: Apply negative exponent rule $x^{-n}=\frac{1}{x^n}$

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

Step3: Distribute exponent to factors

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

Step4: Simplify each factor

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

Step5: Combine results

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

---

Step1: Apply power rule $(x^m)^n=x^{m \cdot n}$

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

Step2: Calculate exponent product

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

---

Step1: Convert decimal exponent to fraction

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

Step2: Distribute exponent to factors

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

Step3: Simplify each factor

$81^{\frac{5}{4}}=(3^4)^{\frac{5}{4}}=3^5=243$, $(x^{12})^{\frac{5}{4}}=x^{12 \cdot \frac{5}{4}}=x^{15}$

Step4: Combine results

$243x^{15}$

Answer:

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