Question

rational exponents
exponent
on
radicand
\\(\sqrt5{7^2} = 7^{\frac{2}{5}}\\)
root/
index
basic rules for exp
product rule
\\(a^m \cdot a^n = a^{m + n}\\)
31)
write \\(\frac{x\sqrt{x^3}}{\sqrt3{x^5}}\\) as a single term in simplest form, with a rational exponent.
which expression is an equivalent form of \\(a\sqrt5{a^4}\\)?
(1) \\(a\\)
(3) \\(a^{\frac{9}{4}}\\)
(4) \\(a^{\frac{1}{5}}\\)

Explanation:

First Problem (Simplify \(\frac{x\sqrt{x^3}}{\sqrt[3]{x^5}}\))

Step1: Convert radicals to exponents

Recall that \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\) and \(a = a^1\). So, \(\sqrt{x^3}=x^{\frac{3}{2}}\) and \(\sqrt[3]{x^5}=x^{\frac{5}{3}}\), and \(x = x^1\). The expression becomes \(\frac{x^1 \cdot x^{\frac{3}{2}}}{x^{\frac{5}{3}}}\).

Step2: Use product rule for exponents (numerator)

The product rule for exponents is \(a^m \cdot a^n = a^{m + n}\). For the numerator \(x^1 \cdot x^{\frac{3}{2}}\), we add the exponents: \(1+\frac{3}{2}=\frac{2}{2}+\frac{3}{2}=\frac{5}{2}\). So the numerator is \(x^{\frac{5}{2}}\), and the expression is now \(\frac{x^{\frac{5}{2}}}{x^{\frac{5}{3}}}\).

Step3: Use quotient rule for exponents

The quotient rule for exponents is \(\frac{a^m}{a^n}=a^{m - n}\). Subtract the exponents: \(\frac{5}{2}-\frac{5}{3}=\frac{15}{6}-\frac{10}{6}=\frac{5}{6}\). So the simplified form is \(x^{\frac{5}{6}}\).

Second Problem (Equivalent form of \(a\sqrt[5]{a^4}\))

Step1: Convert radical to exponent

Using \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), \(\sqrt[5]{a^4}=a^{\frac{4}{5}}\). So the expression \(a\sqrt[5]{a^4}\) becomes \(a^1 \cdot a^{\frac{4}{5}}\).

Step2: Use product rule for exponents

Add the exponents: \(1+\frac{4}{5}=\frac{5}{5}+\frac{4}{5}=\frac{9}{5}\). So the equivalent form is \(a^{\frac{9}{5}}\).

Answer:

s:

• For \(\frac{x\sqrt{x^3}}{\sqrt[3]{x^5}}\): \(x^{\frac{5}{6}}\)
• For \(a\sqrt[5]{a^4}\): The equivalent form is \(a^{\frac{9}{5}}\) (assuming the option with \(\frac{9}{5}\) exponent for \(a\) is the correct one, likely option (2) if we follow the partial marking, but based on calculation, the exponent is \(\frac{9}{5}\))