The expression root(6)(10^(5))*root(3)(10^(2)) is equivalent to 10^((5)/(9)) 100^((5)/(9)) 10^((3)/(2)) 100^((3)/(2))

Rewrite Roots as Exponents: Rewrite the roots as fractional exponents. The 6th root of 10^5 can be written as (10^5)^(1/6), and the cube root of 10^2 can be written as (10^2)^(1/3).Apply Power of Power Rule: Apply the power of a power rule. When you have a power to a power, you multiply the exponents. So, (10^5)^(1/6) becomes 10^(5/6), and (10^2)^(1/3) becomes 10^(2/3).Multiply Expressions: Multiply the two expressions. Now we have 10^(5/6) * 10^(2/3). Since the bases are the same, we can add the exponents.Add Exponents: Add the exponents. We add the fractions 5/6 and 2/3. To add these fractions, we need a common denominator, which is 6. So we convert 2/3 to 4/6 and then add: (5/6) + (4/6) = (9/6).Simplify Exponent: Simplify the exponent. The fraction 9/6 can be simplified to 3/2. So now we have 10^(3/2).Check for Further Simplification: Determine if the expression can be further simplified. 10^(3/2) is already in its simplest form, and it matches one of the answer choices.

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The expression
root(6)(10^(5))*root(3)(10^(2)) is equivalent to

10^((5)/(9))

100^((5)/(9))

10^((3)/(2))

100^((3)/(2))

The expression $\sqrt[6]{10^{5}} \cdot \sqrt[3]{10^{2}}$ is equivalent to $\newline$ $10^{\frac{5}{9}}$ $\newline$ $100^{\frac{5}{9}}$ $\newline$ $10^{\frac{3}{2}}$ $\newline$ $100^{\frac{3}{2}}$

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Math Problems

Precalculus

Operations with rational exponents

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Q. The expression

\sqrt[6]{10^{5}} \cdot \sqrt[3]{10^{2}}

is equivalent to

\newline

10^{\frac{5}{9}}

\newline

100^{\frac{5}{9}}

\newline

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

\newline

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

Rewrite Roots as Exponents: Rewrite the roots as fractional exponents. $\newline$ The $6^{\text{th}}$ root of $10^5$ can be written as $(10^5)^{\frac{1}{6}}$ , and the cube root of $10^2$ can be written as $(10^2)^{\frac{1}{3}}$ .
Apply Power of Power Rule: Apply the power of a power rule. $\newline$ When you have a power to a power, you multiply the exponents. So, $(10^5)^{1/6}$ becomes $10^{5/6}$ , and $(10^2)^{1/3}$ becomes $10^{2/3}$ .
Multiply Expressions: Multiply the two expressions. $\newline$ Now we have $10^{\frac{5}{6}} \times 10^{\frac{2}{3}}$ . Since the bases are the same, we can add the exponents.
Add Exponents: Add the exponents. $\newline$ We add the fractions $\frac{5}{6}$ and $\frac{2}{3}$ . To add these fractions, we need a common denominator, which is $6$ . So we convert $\frac{2}{3}$ to $\frac{4}{6}$ and then add: $\left(\frac{5}{6}\right) + \left(\frac{4}{6}\right) = \left(\frac{9}{6}\right)$ .
Simplify Exponent: Simplify the exponent. $\newline$ The fraction $\frac{9}{6}$ can be simplified to $\frac{3}{2}$ . So now we have $10^{\frac{3}{2}}$ .
Check for Further Simplification: Determine if the expression can be further simplified. $10^{\frac{3}{2}}$ is already in its simplest form, and it matches one of the answer choices.