The expression root(6)(3^(5))*root(5)(3^(4)) is equivalent to 9^((2)/(3)) 9^((49)/(30)) 3^((49)/(30)) 3^((2)/(3))

Rewrite using radical notation: Understand the given expression and rewrite it using radical notation. The given expression is root(6)(3^(5))*root(5)(3^(4)). This can be rewritten using exponents as: (3^(5))^(1/6) * (3^(4))^(1/5)Apply power of a power rule: Apply the power of a power rule. According to the power of a power rule, (a^m)^n = a^(m*n). We apply this rule to both terms: (3^(5*1/6)) * (3^(4*1/5))Perform exponent multiplication: Perform the multiplication of the exponents. Now we multiply the exponents for each term: 3^(5/6) * 3^(4/5)Add exponents with same base: Add the exponents since the bases are the same. When multiplying with the same base, we add the exponents: 3^((5/6) + (4/5))Find common denominator: Find a common denominator to add the fractions. The common denominator for 6 and 5 is 30. We convert each fraction to have a denominator of 30: 3^((25/30) + (24/30))Add fractions: Add the fractions. Now we add the numerators of the fractions: 3^((25 + 24)/30) 3^((49)/30)Check for further simplification: Check if the expression can be simplified further. The expression 3^((49)/30) is already in its simplest form, so no further simplification is needed.

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The expression
root(6)(3^(5))*root(5)(3^(4)) is equivalent to

9^((2)/(3))

9^((49)/(30))

3^((49)/(30))

3^((2)/(3))

The expression $\sqrt[6]{3^{5}} \cdot \sqrt[5]{3^{4}}$ is equivalent to $\newline$ $9^{\frac{2}{3}}$ $\newline$ $9^{\frac{49}{30}}$ $\newline$ $3^{\frac{49}{30}}$ $\newline$ $3^{\frac{2}{3}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

\sqrt[6]{3^{5}} \cdot \sqrt[5]{3^{4}}

is equivalent to

\newline

9^{\frac{2}{3}}

\newline

9^{\frac{49}{30}}

\newline

3^{\frac{49}{30}}

\newline

3^{\frac{2}{3}}

Rewrite using radical notation: Understand the given expression and rewrite it using radical notation. $\newline$ The given expression is $\sqrt[6]{3^{5}}$ $\cdot$ $\sqrt[5]{3^{4}}$ . This can be rewritten using exponents as: $\newline$ $(3^{5})^{\frac{1}{6}} \cdot (3^{4})^{\frac{1}{5}}$
Apply power of a power rule: Apply the power of a power rule. $\newline$ According to the power of a power rule, $(a^m)^n = a^{m*n}$ . We apply this rule to both terms: $\newline$ $(3^{5*1/6}) * (3^{4*1/5})$
Perform exponent multiplication: Perform the multiplication of the exponents. $\newline$ Now we multiply the exponents for each term: $\newline$ $3^{\frac{5}{6}} \times 3^{\frac{4}{5}}$
Add exponents with same base: Add the exponents since the bases are the same. $\newline$ When multiplying with the same base, we add the exponents: $\newline$ $3^{(\frac{5}{6} + \frac{4}{5})}$
Find common denominator: Find a common denominator to add the fractions. $\newline$ The common denominator for $6$ and $5$ is $30$ . We convert each fraction to have a denominator of $30$ : $\newline$ $3^{\left(\frac{25}{30} + \frac{24}{30}\right)}$
Add fractions: Add the fractions. $\newline$ Now we add the numerators of the fractions: $\newline$ $3^{\left(\frac{25 + 24}{30}\right)}$ $\newline$ $3^{\left(\frac{49}{30}\right)}$
Check for further simplification: Check if the expression can be simplified further. $\newline$ The expression $3^{(\frac{49}{30})}$ is already in its simplest form, so no further simplification is needed.