Given x > 0, the expression sqrt(x^(19)) is equivalent to x^(8)sqrt(x^(2)) x^(8)sqrtx x^(9)sqrtx x^(9)sqrt(x^(2))

Express in Exponential Form: We start by expressing the square root of x to the power of 19 in exponential form. sqrt(x^(19)) = (x^(19))^(1/2)Apply Power to Power Rule: Next, we apply the power to power rule of exponents, which states that (a^m)^n = a^(m * n). (x^(19))^(1/2) = x^(19/2)Simplify the Exponent: Now, we need to simplify the exponent. The fraction 19/2 can be rewritten as 9 + 1/2, since 19/2 = 18/2 + 1/2 = 9 + 1/2. x^(19/2) = x^(9 + 1/2)Split Exponent into Two Parts: We can now split the exponent into two parts using the property of exponents that states a^(m + n) = a^m * a^n. x^(9 + 1/2) = x^9 * x^(1/2)Recognize Square Root: Finally, we recognize that x^(1/2) is the square root of x. x^9 * x^(1/2) = x^9 * sqrt(x)

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Given
x > 0, the expression
sqrt(x^(19)) is equivalent to

x^(8)sqrt(x^(2))

x^(8)sqrtx

x^(9)sqrtx

x^(9)sqrt(x^(2))

Given x>0 , the expression $\sqrt{x^{19}}$ is equivalent to $\newline$ $x^{8} \sqrt{x^{2}}$ $\newline$ $x^{8} \sqrt{x}$ $\newline$ $x^{9} \sqrt{x}$ $\newline$ $x^{9} \sqrt{x^{2}}$

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Math Problems

Algebra 2

Simplify radical expressions with root inside the root

Full solution

Q. Given

x>0

, the expression

\sqrt{x^{19}}

is equivalent to

\newline

x^{8} \sqrt{x^{2}}

\newline

x^{8} \sqrt{x}

\newline

x^{9} \sqrt{x}

\newline

x^{9} \sqrt{x^{2}}

Express in Exponential Form: We start by expressing the square root of $x$ to the power of $19$ in exponential form. $\sqrt{x^{19}} = (x^{19})^{\frac{1}{2}}$
Apply Power to Power Rule: Next, we apply the power to power rule of exponents, which states that $(a^m)^n = a^{(m \cdot n)}$ . $\newline$ $(x^{19})^{\frac{1}{2}} = x^{\frac{19}{2}}$
Simplify the Exponent: Now, we need to simplify the exponent. The fraction $\frac{19}{2}$ can be rewritten as $9 + \frac{1}{2}$ , since $\frac{19}{2} = \frac{18}{2} + \frac{1}{2} = 9 + \frac{1}{2}$ . $\newline$ $x^{\frac{19}{2}} = x^{9 + \frac{1}{2}}$
Split Exponent into Two Parts: We can now split the exponent into two parts using the property of exponents that states $a^{(m + n)} = a^m \cdot a^n$ . $\newline$ $x^{(9 + \frac{1}{2})} = x^9 \cdot x^{\frac{1}{2}}$
Recognize Square Root: Finally, we recognize that $x^{(1/2)}$ is the square root of $x$ . $\newline$ $x^9 \cdot x^{(1/2)} = x^9 \cdot \sqrt{x}$