Use exponent laws to simplify $(\sqrt[8]{x})\left(\sqrt[5]{x^{3}}\right)$ .

Full solution

Q. Use exponent laws to simplify

(\sqrt[8]{x})\left(\sqrt[5]{x^{3}}\right)

Rewrite Roots as Exponents: Rewrite the roots as fractional exponents. $\newline$ The eighth root of $x$ can be written as $x^{\frac{1}{8}}$ , and the fifth root of $x^3$ can be written as $(x^3)^{\frac{1}{5}}$ .
Apply Power of a Power Rule: Apply the power of a power rule to the second term. $\newline$ When you have a power raised to another power, you multiply the exponents. So, $(x^3)^{\frac{1}{5}}$ becomes $x^{3\cdot\left(\frac{1}{5}\right)}$ which simplifies to $x^{\frac{3}{5}}$ .
Multiply Expressions with Same Base: Multiply the two expressions with the same base. $\newline$ Now we have $x^{1/8} \times x^{3/5}$ . When multiplying with the same base, we add the exponents: $x^{1/8 + 3/5}$ .
Find Common Denominator: Find a common denominator to add the exponents. $\newline$ The common denominator for $8$ and $5$ is $40$ . So we convert the fractions: $(1/8)$ becomes $(5/40)$ and $(3/5)$ becomes $(24/40)$ .
Add Exponents: Add the exponents. $\newline$ Now we add the fractions: $x^{(5/40 + 24/40)}$ which simplifies to $x^{29/40}$ .
Write Final Expression: Write the final simplified expression. $\newline$ The expression is now simplified to $x^{\frac{29}{40}}$ , which is the same as the $40$ th root of $x$ raised to the $29$ th power.