if $3^a=5\sqrt{3^4}$ ,what is the value of $a$ ?

View step-by-step help

Home

Math Problems

Grade 8

Solve equations with variable exponents

Full solution

Q. if

3^a=5\sqrt{3^4}

,what is the value of

a

Identify Given Equation and Goal: Identify the given equation and the goal. $\newline$ We are given that $3^a = 5\sqrt{3^4}$ , and we need to find the value of $a$ .
Rewrite Square Root as Exponent: Rewrite the square root of $3^4$ as an exponent. $\newline$ The square root of a number is the same as raising that number to the power of $1/2$ . So, $\sqrt{3^4}$ is the same as $(3^4)^{1/2}$ .
Apply Power of Power Rule: Apply the power of a power rule. $\newline$ According to the power of a power rule, $(x^m)^n = x^{(m*n)}$ . Therefore, $(3^4)^{1/2} = 3^{(4*(1/2))}$ .
Simplify the Exponent: Simplify the exponent. $3^{4*(1/2)}$ simplifies to $3^2$ because $4*(1/2)$ equals $2$ .
Substitute Simplified Exponent: Substitute the simplified exponent back into the equation. $\newline$ Now we have $3^a = 5 \times 3^2$ .
Divide to Isolate $3^a$ : Divide both sides of the equation by $3^2$ to isolate $3^a$ . Doing this, we get $(3^a) / (3^2) = 5$ .
Apply Quotient of Powers Rule: Apply the quotient of powers rule. $\newline$ According to the quotient of powers rule, $a^m / a^n = a^{m-n}$ . Therefore, $(3^a) / (3^2) = 3^{a-2}$ .
Set Expression Equal to $5$ : Set the expression equal to $5$ . Now we have $3^{(a-2)} = 5$ .
Solve for $a$ with Logarithm: Solve for $a$ by taking the logarithm of both sides. $\newline$ To solve for $a$ , we can take the logarithm with base $3$ of both sides, which gives us $a - 2 = \log_3(5)$ .
Isolate $a$ by Adding $2$ : Isolate $a$ by adding $2$ to both sides. $\newline$ Adding $2$ to both sides gives us $a = \log_3(5) + 2$ .
Calculate $\log_3(5)$ : Calculate the value of $\log_3(5)$ . Using a calculator or logarithm properties, we find that $\log_3(5)$ is approximately $1.465$ (rounded to three decimal places).
Add $2$ to $\log_3(5)$ : Add $2$ to the approximate value of $\log_3(5)$ . Adding $2$ to $1.465$ gives us $a \approx 1.465 + 2$ .
Calculate Final Approximate Value: Calculate the final approximate value of $a$ . $a \approx 1.465 + 2 \approx 3.465$ .