Question

(1/4)^x * (64/25)^x = (5/4)^2

Answer 1

$(\frac{1}{4})^x * ( \frac{64}{25} )^x = ( \frac{5}{4} )^2 \\ ( \frac{1}{4}* \frac{64}{25} )^x = \frac{25}{16} \\ ( \frac{25}{16} )^x = (\frac{25}{16})^1 \\ x = 1$[/tex]

Answer 2

To solve the equation $(1/4)^x * (64/25)^x = (5/4)^2$, you can simplify the terms on both sides of the equation and then solve for $x$. Here's how you can do it step by step:

First, rewrite the fractions with common bases: $(1/4)^x * (64/25)^x = (5/4)^2$

Now, express the fractions with the same base. Since $64/25$ is greater than 1, you can rewrite it as $25/64$ and then invert it to get the base of $4/5$: $(1/4)^x * (25/64)^x = (5/4)^2$

Now, rewrite the equation with the common base of $4/5$: $(4/5)^x * (25/64)^x = (5/4)^2$

Next, simplify the exponents using the properties of exponents. When you multiply two numbers with the same base, you can add their exponents: $(4/5)^{x + 2x} = (5/4)^2$

Now, simplify the exponents: $(4/5)^{3x} = (5/4)^2$

Now, we have the equation in a simplified form: $(4/5)^{3x} = (5/4)^2$

To solve for $x$, take the logarithm of both sides to eliminate the exponent: $\log((4/5)^{3x}) = \log((5/4)^2)$

Now, you can use the properties of logarithms to bring down the exponents: $3x \cdot \log(4/5) = 2 \cdot \log(5/4)$

Now, isolate $x$ by dividing both sides by $3 \cdot \log(4/5)$: $x = \frac{2 \cdot \log(5/4)}{3 \cdot \log(4/5)}$

You can calculate the approximate value of $x$ using a calculator: $x \approx \frac{2 \cdot \log(5/4)}{3 \cdot \log(4/5)} \approx 1.096$

So, the approximate value of $x$ is approximately 1.096.