Question

no calculator is allowed on this question.
solve the following equation.
$3^{2x} \cdot 9^{4x + 1} = 81$
select one answer
a $x = \frac{1}{8}$
b $x = \frac{1}{5}$
c $x = \frac{7}{8}$
d $x = \frac{5}{2}$

Explanation:

Step1: Rewrite bases as powers of 3

We know that \(9 = 3^2\) and \(81=3^4\). So rewrite the equation \(3^{2x}\cdot9^{4x + 1}=81\) as:
\(3^{2x}\cdot(3^2)^{4x + 1}=3^4\)

Step2: Simplify exponents using power - of - a - power rule

Using the power - of - a - power rule \((a^m)^n=a^{mn}\), we have \((3^2)^{4x + 1}=3^{2(4x + 1)}=3^{8x+2}\).
The equation becomes \(3^{2x}\cdot3^{8x + 2}=3^4\)

Step3: Use the product rule of exponents

Using the product rule of exponents \(a^m\cdot a^n=a^{m + n}\), we get \(3^{2x+(8x + 2)}=3^4\), which simplifies to \(3^{10x+2}=3^4\)

Step4: Set exponents equal (since bases are equal)

If \(a^m=a^n\) and \(a>0,a
eq1\), then \(m = n\). So we set \(10x+2 = 4\)

Step5: Solve for x

Subtract 2 from both sides: \(10x=4 - 2=2\)
Divide both sides by 10: \(x=\frac{2}{10}=\frac{1}{5}\) (Wait, this is wrong. Let's re - do step 2 - step 5)

Wait, let's start over:

Step1: Rewrite bases as powers of 3

The original equation is \(3^{2x}\cdot9^{4x + 1}=81\)
Since \(9 = 3^2\) and \(81 = 3^4\), substitute these values into the equation:
\(3^{2x}\cdot(3^2)^{4x + 1}=3^4\)

Step2: Simplify the left - hand side exponents

Using \((a^m)^n=a^{mn}\), \((3^2)^{4x + 1}=3^{2\times(4x + 1)}=3^{8x+2}\)
Then \(3^{2x}\cdot3^{8x + 2}=3^{2x+(8x + 2)}\) (by \(a^m\cdot a^n=a^{m + n}\))
\(3^{2x+(8x + 2)}=3^{10x + 2}\)
So the equation is \(3^{10x+2}=3^4\)

Step3: Equate the exponents

Since the bases are the same (\(a = 3>0,a
eq1\)), we have \(10x+2 = 4\)
Subtract 2 from both sides: \(10x=4 - 2=2\)
Divide both sides by 10: \(x=\frac{2}{10}=\frac{1}{5}\)? Wait, but let's check the options. Option B is \(x = \frac{1}{5}\)

Wait, maybe I made a mistake in the first re - check. Let's verify:

If \(x=\frac{1}{5}\), then:

Left - hand side: \(3^{2\times\frac{1}{5}}\cdot9^{4\times\frac{1}{5}+1}=3^{\frac{2}{5}}\cdot9^{\frac{4 + 5}{5}}=3^{\frac{2}{5}}\cdot9^{\frac{9}{5}}\)

Since \(9 = 3^2\), \(9^{\frac{9}{5}}=(3^2)^{\frac{9}{5}}=3^{\frac{18}{5}}\)

Then \(3^{\frac{2}{5}}\cdot3^{\frac{18}{5}}=3^{\frac{2 + 18}{5}}=3^{\frac{20}{5}}=3^4 = 81\), which matches the right - hand side.

Wait, but the options are A. \(x=\frac{1}{8}\), B. \(x=\frac{1}{5}\), C. \(x=\frac{7}{8}\), D. \(x=\frac{5}{2}\)

So the correct value of \(x\) is \(\frac{1}{5}\)

Answer:

B. \(x=\frac{1}{5}\)