Question

es of a function such as domain, range, independent, and dependent variableing expressions. do not leave negative exponents. multiply out2.) \\(\frac{10^{2}r^{1}s}{s^{-4}}\\)4.) \\(\sqrt3{250n}\\)6.) simplify the following expression:\\((64p^{2}q^{3})^{\frac{2}{3}}\\)

Explanation:

Step1: Simplify $10^2$ and handle exponents

$10^2 = 100$; use $\frac{s^a}{s^b}=s^{a-b}$:
$\frac{10^2 r^1 s^1}{s^{-4}} = 100 r s^{1-(-4)}$

Step2: Calculate the exponent of $s$

$1-(-4)=1+4=5$:
$100 r s^{5}$

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Step1: Factor 250 into perfect cube

$250=125\times2=5^3\times2$:
$\sqrt[3]{250n}=\sqrt[3]{5^3\times2n}$

Step2: Extract perfect cube factor

Use $\sqrt[3]{ab}=\sqrt[3]{a}\sqrt[3]{b}$:
$5\sqrt[3]{2n}$

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Step1: Apply power rule to each term

Use $(ab)^c=a^c b^c$:
$(64p^2 q^3)^{\frac{2}{3}}=64^{\frac{2}{3}} (p^2)^{\frac{2}{3}} (q^3)^{\frac{2}{3}}$

Step2: Simplify each base's exponent

$64^{\frac{2}{3}}=(4^3)^{\frac{2}{3}}=4^2=16$; $(p^2)^{\frac{2}{3}}=p^{\frac{4}{3}}$; $(q^3)^{\frac{2}{3}}=q^2$:
$16 p^{\frac{4}{3}} q^2$

Step3: Rewrite $p^{\frac{4}{3}}$ without fraction

$p^{\frac{4}{3}}=p p^{\frac{1}{3}}=p\sqrt[3]{p}$:
$16 p q^2 \sqrt[3]{p}$

Answer:

2.) $100 r s^5$
4.) $5\sqrt[3]{2n}$
6.) $16 p q^2 \sqrt[3]{p}$ (or $16 p^{\frac{4}{3}} q^2$)