Determine The Value Of The Variable In The Simplified Expressions.1. What Is The Value Of $x$? $\[ \begin{array}{l} G^x + H^{-3} = \frac{1}{g^6} + \frac{1}{h^3} \\ X = \square \end{array} \\]2. What Is The Value Of

Introduction

In mathematics, solving for the value of a variable in simplified expressions is a crucial skill that helps us understand and manipulate algebraic equations. In this article, we will explore how to determine the value of a variable in a simplified expression using a step-by-step approach. We will use two examples to illustrate this concept.

Example 1: Solving for x

Problem Statement

What is the value of $x$?

{ \begin{array}{l} g^x + h^{-3} = \frac{1}{g^6} + \frac{1}{h^3} \\ x = \square \end{array} \}

Step 1: Simplify the Right-Hand Side of the Equation

To solve for $x$, we need to simplify the right-hand side of the equation. We can start by finding a common denominator for the two fractions.

$${ \frac{1}{g^6} + \frac{1}{h^3} = \frac{h^3 + g^6}{g^6h^3} }$$

Step 2: Rewrite the Equation with the Simplified Right-Hand Side

Now that we have simplified the right-hand side of the equation, we can rewrite the equation as follows:

$${ g^x + h^{-3} = \frac{h^3 + g^6}{g^6h^3} }$$

Step 3: Multiply Both Sides of the Equation by $g^6h^3$

To eliminate the fraction, we can multiply both sides of the equation by $g^6h^3$.

$${ g^6h^3(g^x + h^{-3}) = g^6h^3 \cdot \frac{h^3 + g^6}{g^6h^3} }$$

Step 4: Simplify the Left-Hand Side of the Equation

Now that we have multiplied both sides of the equation by $g^6h^3$, we can simplify the left-hand side of the equation.

$${ g^{x+6}h^3 + g^6h^{-3} = h^3 + g^6 }$$

Step 5: Subtract $h^3$ from Both Sides of the Equation

To isolate the term with $x$, we can subtract $h^3$ from both sides of the equation.

$${ g^{x+6}h^3 - h^3 + g^6h^{-3} = g^6 }$$

Step 6: Factor Out $h^3$ from the First Term

Now that we have subtracted $h^3$ from both sides of the equation, we can factor out $h^3$ from the first term.

$${ h^3(g^{x+6} - 1) + g^6h^{-3} = g^6 }$$

Step 7: Subtract $g^6h^{-3}$ from Both Sides of the Equation

To isolate the term with $x$, we can subtract $g^6h^{-3}$ from both sides of the equation.

$${ h^3(g^{x+6} - 1) = g^6 - g^6h^{-3} }$$

Step 8: Divide Both Sides of the Equation by $h^3$

To solve for $x$, we can divide both sides of the equation by $h^3$.

$${ g^{x+6} - 1 = \frac{g^6 - g^6h^{-3}}{h^3} }$$

Step 9: Simplify the Right-Hand Side of the Equation

Now that we have divided both sides of the equation by $h^3$, we can simplify the right-hand side of the equation.

$${ g^{x+6} - 1 = \frac{g^6(1 - h^{-3})}{h^3} }$$

Step 10: Add 1 to Both Sides of the Equation

To isolate the term with $x$, we can add 1 to both sides of the equation.

$${ g^{x+6} = 1 + \frac{g^6(1 - h^{-3})}{h^3} }$$

Step 11: Take the Logarithm of Both Sides of the Equation

To solve for $x$, we can take the logarithm of both sides of the equation.

$${ \log(g^{x+6}) = \log\left(1 + \frac{g^6(1 - h^{-3})}{h^3}\right) }$$

Step 12: Use the Power Rule of Logarithms

Now that we have taken the logarithm of both sides of the equation, we can use the power rule of logarithms to simplify the left-hand side of the equation.

$${ (x+6)\log(g) = \log\left(1 + \frac{g^6(1 - h^{-3})}{h^3}\right) }$$

Step 13: Divide Both Sides of the Equation by $\log(g)$

To solve for $x$, we can divide both sides of the equation by $\log(g)$.

$${ x+6 = \frac{\log\left(1 + \frac{g^6(1 - h^{-3})}{h^3}\right)}{\log(g)} }$$

Step 14: Subtract 6 from Both Sides of the Equation

To isolate the value of $x$, we can subtract 6 from both sides of the equation.

$${ x = \frac{\log\left(1 + \frac{g^6(1 - h^{-3})}{h^3}\right)}{\log(g)} - 6 }$$

Example 2: Solving for y

Problem Statement

What is the value of $y$?

{ \begin{array}{l} \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2} + \frac{1}{d^2} \\ y = \square \end{array} \}

Step 1: Simplify the Right-Hand Side of the Equation

To solve for $y$, we need to simplify the right-hand side of the equation. We can start by finding a common denominator for the two fractions.

$${ \frac{1}{c^2} + \frac{1}{d^2} = \frac{d^2 + c^2}{c^2d^2} }$$

Step 2: Rewrite the Equation with the Simplified Right-Hand Side

Now that we have simplified the right-hand side of the equation, we can rewrite the equation as follows:

$${ \frac{1}{a^2} + \frac{1}{b^2} = \frac{d^2 + c^2}{c^2d^2} }$$

Step 3: Multiply Both Sides of the Equation by $c^2d^2$

To eliminate the fraction, we can multiply both sides of the equation by $c^2d^2$.

$${ c^2d^2\left(\frac{1}{a^2} + \frac{1}{b^2}\right) = c^2d^2 \cdot \frac{d^2 + c^2}{c^2d^2} }$$

Step 4: Simplify the Left-Hand Side of the Equation

Now that we have multiplied both sides of the equation by $c^2d^2$, we can simplify the left-hand side of the equation.

$${ c^2d^2\left(\frac{1}{a^2} + \frac{1}{b^2}\right) = d^2 + c^2 }$$

Step 5: Subtract $d^2$ from Both Sides of the Equation

To isolate the term with $y$, we can subtract $d^2$ from both sides of the equation.

$${ c^2d^2\left(\frac{1}{a^2} + \frac{1}{b^2}\right) - d^2 = c^2 }$$

Step 6: Factor Out $d^2$ from the First Term

Now that we have subtracted $d^2$ from both sides of the equation, we can factor out $d^2$ from the first term.

$${ d^2\left(c^2\left(\frac{1}{a^2} + \frac{1}{b^2}\right) - 1\right) = c^2 }$$

Step 7: Divide Both Sides of the Equation by $d^2$

To solve for $y$, we can divide both sides of the equation by $d^2$.

$${ c^2\left(\frac{1}{a^2} + \frac{1}{b^2}\right) - 1 = \frac{c^2}{d^2} }$$

Step 8: Add 1 to Both Sides of the Equation

To isolate the term with $y$, we can add 1 to both sides of the equation.

$[ c^{2\left(\frac{1}{a}2} + \frac{1}{b^2}\right) =