The expression root(6)(2^(5))*root(3)(2^(5)) is equivalent to 2^((5)/(2)) 2^((2)/(5)) 2^((25)/(18)) 2^((18)/(25))

Understand the expression: Understand the given expression. We have the expression root(6)(2^(5))*root(3)(2^(5)), which involves two different roots of the same base raised to the same power.Convert roots to exponents: Convert roots to fractional exponents. The 6th root of 2^5 can be written as (2^5)^(1/6), and the cube root of 2^5 can be written as (2^5)^(1/3). So, the expression becomes (2^5)^(1/6) * (2^5)^(1/3).Apply power of a power rule: Apply the power of a power rule. When you have a power of a power, you multiply the exponents. So, (2^5)^(1/6) becomes 2^(5/6) and (2^5)^(1/3) becomes 2^(5/3). Now the expression is 2^(5/6) * 2^(5/3).Add exponents when multiplying: Add the exponents when multiplying with the same base. According to the laws of exponents, when you multiply terms with the same base, you add the exponents. So, we add 5/6 and 5/3.Find common denominator: Find a common denominator and add the exponents. The common denominator for 6 and 3 is 6. So, we convert 5/3 to 10/6. Now we have 2^(5/6) * 2^(10/6). Adding the exponents gives us 2^((5/6) + (10/6)) = 2^(15/6).Simplify the exponent: Simplify the exponent. 15/6 simplifies to 5/2 because 15 divided by 3 is 5, and 6 divided by 3 is 2. So, the expression simplifies to 2^(5/2).

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The expression
root(6)(2^(5))*root(3)(2^(5)) is equivalent to

2^((5)/(2))

2^((2)/(5))

2^((25)/(18))

2^((18)/(25))

The expression $\sqrt[6]{2^{5}} \cdot \sqrt[3]{2^{5}}$ is equivalent to $\newline$ $2^{\frac{5}{2}}$ $\newline$ $2^{\frac{2}{5}}$ $\newline$ $2^{\frac{25}{18}}$ $\newline$ $2^{\frac{18}{25}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

\sqrt[6]{2^{5}} \cdot \sqrt[3]{2^{5}}

is equivalent to

\newline

2^{\frac{5}{2}}

\newline

2^{\frac{2}{5}}

\newline

2^{\frac{25}{18}}

\newline

2^{\frac{18}{25}}

Understand the expression: Understand the given expression. $\newline$ We have the expression $\sqrt[6]{2^{5}}$ $\cdot$ $\sqrt[3]{2^{5}}$ , which involves two different roots of the same base raised to the same power.
Convert roots to exponents: Convert roots to fractional exponents. $\newline$ The $6^{\text{th}}$ root of $2^5$ can be written as $(2^5)^{\frac{1}{6}}$ , and the cube root of $2^5$ can be written as $(2^5)^{\frac{1}{3}}$ . $\newline$ So, the expression becomes $(2^5)^{\frac{1}{6}} * (2^5)^{\frac{1}{3}}$ .
Apply power of a power rule: Apply the power of a power rule. $\newline$ When you have a power of a power, you multiply the exponents. So, $(2^5)^{1/6}$ becomes $2^{5/6}$ and $(2^5)^{1/3}$ becomes $2^{5/3}$ . $\newline$ Now the expression is $2^{5/6} \times 2^{5/3}$ .
Add exponents when multiplying: Add the exponents when multiplying with the same base. $\newline$ According to the laws of exponents, when you multiply terms with the same base, you add the exponents. So, we add $\frac{5}{6}$ and $\frac{5}{3}$ .
Find common denominator: Find a common denominator and add the exponents. $\newline$ The common denominator for $6$ and $3$ is $6$ . So, we convert $\frac{5}{3}$ to $\frac{10}{6}$ . Now we have $2^{\frac{5}{6}} \times 2^{\frac{10}{6}}$ . $\newline$ Adding the exponents gives us $2^{(\frac{5}{6} + \frac{10}{6})} = 2^{\frac{15}{6}}$ .
Simplify the exponent: Simplify the exponent. $\frac{15}{6}$ simplifies to $\frac{5}{2}$ because $15$ divided by $3$ is $5$ , and $6$ divided by $3$ is $2$ . So, the expression simplifies to $2^{\frac{5}{2}}$ .