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Understanding rational exponents is crucial for solving equations in algebra. A rational exponent, such as the one found in the equation \( (x-7)^{2/3} = 9 \), represents a power that is expressed as a fraction. The number on the top of the fraction denotes the power (also known as the numerator), and the number below (the denominator) represents the root. In this case, \(2/3\) implies that \(x-7\) should be squared (the numerator), and then the cube root (the denominator) should be taken.

To remove a rational exponent and simplify the equation, you can use the property of exponents that states raising a power to a power multiplies the exponents. Therefore, to get rid of the denominator of the rational exponent, you raise both sides of the equation to the reciprocal power, in our example, to the power of 3. This inverses the cube root and leaves you with \( (x-7)^2 \), which is a much simpler equation to work with. Understanding the logic behind this step is vital to mastering algebraic solutions with rational exponents.

When solving equations with rational exponents, the goal is to find all possible algebraic solutions—values of the variable that make the equation true. In our example, after removing the cube root by raising each side of \( (x-7)^{2/3} = 9 \) to the power of 3, we obtain a square equation \( (x-7)^2 = 729 \). The next step is to take the square root of both sides, yielding two possible algebraic solutions: \( x-7 = \pm 27 \).

The notation \( \pm \) means that both the positive and negative roots must be considered as potential solutions. By adding 7 to both possible roots, we find two values for \(x\): \( x = 7 + 27 \) and \( x = 7 - 27 \). This process is a classic example of finding algebraic solutions and demonstrates how effectively algebra can unpack and simplify complex problems to find multiple possible answers.

Finding potential solutions is only half the battle when it comes to algebraic equations with rational exponents. It's essential to verify that the solutions are valid by substituting them back into the original equation. This step is often referred to as 'checking' the solutions.

Using our example, the two algebraic solutions for \(x\) are \(x = 34\) and \(x = -20\). However, when we substitute each one back into the original equation \((x-7)^{2/3}=9\), only \( x = 34 \) holds true. The erroneous root, \(x = -20\), when substituted into \( (x-7)^{2/3} \), does not yield 9. It's critical to recognize that not every solution derived algebraically will satisfy the equation—some may be extraneous due to the steps we've taken to solve the equation, such as squaring or square rooting, which can introduce additional solutions that must be discarded. A thorough check ensures that only mathematically valid solutions are considered.