Question

\(2^{x}=4^{y}=8^{z}\) and \(x y z=288\), then value of \(\frac{1}{2 x}+\frac{1}{4 y}+\frac{1}{8 z}\) is (a) \(\frac{11}{12}\) (b) \(\frac{11}{96}\) (c) \(\frac{29}{96}\) (d) none of these

Short answer

Answer: \(\boxed{\frac{11}{96}}\).

Step-by-step solution

Rewrite the first equation using one exponent

We can rewrite the first equation using base 2 exponents: \(2^x = (2^2)^y = (2^3)^z\). Now, we have \(2^x = 2^{2y} = 2^{3z}\).

Solve the first equation for x, y, and z in terms of one variable

Since we now have equal exponents in the first equation, we can equate them: \(x = 2y = 3z\). Divide the first part by 2 and the second part by 3. We will get the expressions for y and z in terms of x: \(y = \frac{1}{2}x\) and \(z = \frac{1}{3}x\).

Solve the second equation for x

Now, substitute the expressions for y and z from step 2 into the second equation, \(xyz = 288\): \(x\left(\frac{1}{2}x\right)\left(\frac{1}{3}x\right) = 288\). Simplify: \(\frac{1}{6}x^3 = 288\). Multiply both sides by 6 to get rid of the fraction: \(x^3 = 1728\). Taking the cube root of both sides, we find that \(x = 12\).

Find the values of y and z

Plug the value of x = 12 back into the expressions for y and z: \(y = \frac{1}{2}(12) = 6\) and \(z = \frac{1}{3}(12) = 4\).

Substitute the values of x, y, and z into the expression

Now, substitute the values of x = 12, y = 6, and z = 4 into the expression: \(\frac{1}{2(12)} + \frac{1}{4(6)} + \frac{1}{8(4)}\). Simplify to find the final value: \(\frac{1}{24} + \frac{1}{24} + \frac{1}{32} = \frac{2}{24} + \frac{1}{32} = \frac{1}{12} + \frac{1}{32} = \frac{11}{96}\). The value of the expression \(\frac{1}{2x} + \frac{1}{4y} + \frac{1}{8z}\) is \(\boxed{\frac{11}{96}}\), which is the answer option (b).

Key concepts

Equations

Equations are mathematical statements that express the equality between two expressions. In the original exercise, we have two primary equations:

• \(2^x = 4^y = 8^z\)
• \(xyz = 288\)

The first is an exponential equation consisting of exponents with different bases.
This means each part of the equation \(2^x, 4^y, 8^z\) relates using a similar pattern, where powers are leveraged to express another.
Equating the exponents helps find a common relationship between the variables \(x, y,\) and \(z\).
The second equation shows a multiplicative relationship, indicating the product of the variables equals \(288\). This gives us specific information needed to solve for each variable, making it possible to substitute and simplify in later steps.

Substitution Method

The substitution method is a way to replace one variable in an equation with an expression involving another variable. In our problem, once the equations are written in terms of the same base (base 2), all the exponents can be correlated:

• \(x = 2y = 3z\)

By expressing \(y\) and \(z\) as fractions of \(x\) (e.g., \(y = \frac{1}{2}x\) and \(z = \frac{1}{3}x\)), substitution can be easily performed in the second equation.
This then allows for transforming complex equations into a simpler form by using quicker arithmetic operations such as multiplication.
Substitution is instrumental in solving simultaneous equations as it efficiently kicks off the process of finding individual variable values.

Simplification

Simplification involves reducing a complex mathematical expression into its simplest form. After substitution, the equation \(x\left(\frac{1}{2}x\right)\left(\frac{1}{3}x\right) = 288\) is simplified as:

• \(\frac{1}{6}x^3 = 288\)

This simplification is achieved by factoring out coefficients and rearranging terms.
The next step involves clearing the fractions by multiplying both sides by \(6\), which results in finding a straightforward cubic equation \(x^3 = 1728\).
Simplifying helps in converting even the most tangled equations into forms that are easy to solve, making sure the solution is both practical and quick.

Cube Roots

To solve equations involving cubic expressions, understanding cube roots is crucial. The cube root is a special instance of powers and roots, specifically the inverse of cubing a number.
In our exercise, once the equation is simplified to \(x^3 = 1728\), we take the cube root of both sides.

• \(x = \sqrt[3]{1728}\)

Calculating the cube root, as it turns out, yields \(x = 12\).
Cube roots allow us to determine the original value that was cubed to reach a specific number.
This concept is vital for breaking down cubic expressions into manageable figures, helping in finding exact values needed to solve the complete problem. This ensures you have the knowledge to deal with equations extending beyond basic arithmetic.