The expression root(6)(10^(5))*root(5)(10^(3)) is equivalent to 10^((43)/(30)) 10^((1)/(2)) 100^((43)/(30)) 100^((1)/(2))

Understand the expression: Understand the given expression. We have the expression root(6)(10^(5)) * root(5)(10^(3)). This means we are multiplying two radical expressions, where the first is the sixth root of 10 raised to the fifth power, and the second is the fifth root of 10 raised to the third power.Convert to fractional exponents: Convert the radical expressions to fractional exponents. The sixth root of 10^5 can be written as (10^5)^(1/6), and the fifth root of 10^3 can be written as (10^3)^(1/5).Apply power of a power rule: Apply the power of a power rule. Using the power of a power rule, we can simplify the expressions further: (10^5)^(1/6) = 10^(5/6) (10^3)^(1/5) = 10^(3/5)Multiply expressions: Multiply the two expressions. Now we multiply the two expressions with the same base (10): 10^(5/6) * 10^(3/5)Apply product of powers rule: Apply the product of powers rule. When multiplying expressions with the same base, we add the exponents: 10^(5/6 + 3/5)Find common denominator: Find a common denominator and add the exponents. The common denominator for 6 and 5 is 30. We convert the fractions to have the same denominator: (5/6) * (5/5) = 25/30 (3/5) * (6/6) = 18/30 Now we add the fractions: 25/30 + 18/30 = 43/30Write final expression: Write the final expression. The final expression after adding the exponents is: 10^(43/30)

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The expression
root(6)(10^(5))*root(5)(10^(3)) is equivalent to

10^((43)/(30))

10^((1)/(2))

100^((43)/(30))

100^((1)/(2))

The expression $\sqrt[6]{10^{5}} \cdot \sqrt[5]{10^{3}}$ is equivalent to $\newline$ $10^{\frac{43}{30}}$ $\newline$ $10^{\frac{1}{2}}$ $\newline$ $100^{\frac{43}{30}}$ $\newline$ $100^{\frac{1}{2}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

\sqrt[6]{10^{5}} \cdot \sqrt[5]{10^{3}}

is equivalent to

\newline

10^{\frac{43}{30}}

\newline

10^{\frac{1}{2}}

\newline

100^{\frac{43}{30}}

\newline

100^{\frac{1}{2}}

Understand the expression: Understand the given expression. $\newline$ We have the expression $\sqrt[6]{10^{5}} \times \sqrt[5]{10^{3}}$ . This means we are multiplying two radical expressions, where the first is the sixth root of $10$ raised to the fifth power, and the second is the fifth root of $10$ raised to the third power.
Convert to fractional exponents: Convert the radical expressions to fractional exponents. $\newline$ The sixth root of $10^5$ can be written as $(10^5)^{\frac{1}{6}}$ , and the fifth root of $10^3$ can be written as $(10^3)^{\frac{1}{5}}$ .
Apply power of a power rule: Apply the power of a power rule. $\newline$ Using the power of a power rule, we can simplify the expressions further: $\newline$ $(10^5)^{1/6} = 10^{5/6}$ $\newline$ $(10^3)^{1/5} = 10^{3/5}$
Multiply expressions: Multiply the two expressions. $\newline$ Now we multiply the two expressions with the same base $10$ : $\newline$ $10^{\frac{5}{6}} \times 10^{\frac{3}{5}}$
Apply product of powers rule: Apply the product of powers rule. $\newline$ When multiplying expressions with the same base, we add the exponents: $\newline$ $10^{\frac{5}{6} + \frac{3}{5}}$
Find common denominator: Find a common denominator and add the exponents. $\newline$ The common denominator for $6$ and $5$ is $30$ . We convert the fractions to have the same denominator: $\newline$ $(\frac{5}{6}) \cdot (\frac{5}{5}) = \frac{25}{30}$ $\newline$ $(\frac{3}{5}) \cdot (\frac{6}{6}) = \frac{18}{30}$ $\newline$ Now we add the fractions: $\newline$ $\frac{25}{30} + \frac{18}{30} = \frac{43}{30}$
Write final expression: Write the final expression. $\newline$ The final expression after adding the exponents is: $\newline$ $10^{\frac{43}{30}}$