Full solution

Q. If

11^a = 4\sqrt{11^3}

, what is the value of

a

Identify Equation Components: Understand the given equation and identify the components. $\newline$ We are given the equation $11^a = 4\sqrt{11^3}$ . The $4\sqrt{}$ symbol represents the fourth root. This means we need to find the value of $a$ such that $11$ raised to the power of $a$ is equal to the fourth root of $11$ cubed.
Express Fourth Root: Express the fourth root in exponent form. $\newline$ The fourth root of a number can be expressed as that number raised to the power of $\frac{1}{4}$ . Therefore, $\sqrt[4]{11^3}$ can be written as $(11^3)^{\frac{1}{4}}$ .
Apply Power Rule: Apply the power of a power rule. $\newline$ According to the power of a power rule, $(a^m)^n = a^{m*n}$ . So, $(11^3)^{1/4}$ can be simplified to $11^{3*(1/4)}$ .
Simplify Exponent: Simplify the exponent. Multiply the exponents to simplify the expression: $11^{3\times(1/4)} = 11^{3/4}.$
Set Equal to Original Equation: Set the simplified expression equal to the original equation. $\newline$ Now we have $11^a = 11^{\frac{3}{4}}$ . Since the bases are the same (both are $11$ ), we can set the exponents equal to each other to find the value of $a$ .
Solve for $a$ : Solve for $a$ . $\newline$ Since $11^a = 11^{\frac{3}{4}}$ , it follows that $a = \frac{3}{4}$ .