91Ó°ÊÓ

Exponents are a way to represent repeated multiplication. For example, in the term \(a^b\), 'a' is the base and 'b' is the exponent, which multiplies 'a' by itself 'b' times. In our problem, we have \((2x - 3)^6\). Here, the entire expression \(2x - 3\) is the base raised to the exponent 6.

This means \(2x - 3\) is multiplied by itself 6 times. Exponents are an important tool in algebra, allowing us to work with large numbers in a more compact manner.

Remember, any non-zero number raised to an exponent results in a positive number. However, if the base is zero, any positive exponent of it will still result in zero. This is why we set our base to zero to solve \((2x - 3)^6 = 0\).

Simplifying polynomials with exponents often involves breaking down high-power terms into their factors.

To solve for a variable means to isolate that variable on one side of the equation. In our example, we started with the equation \((2x - 3)^6 = 0\).

The first step was to remove the exponent since any base raised to a positive power equals zero only when the base itself is zero. Hence, \(2x - 3 = 0\).

Next, we solved for the variable 'x' by isolating it. Here’s how:

• Add 3 to both sides of the equation: \(2x - 3 + 3 = 0 + 3\) simplifies to \(2x = 3\).
• Then, divide both sides by 2: \(2x/2 = 3/2\) simplifies to \(x = 3/2\).

This is a systematic method to isolate the variable.

Always perform the same operation on both sides of the equation to maintain balance.

Verifying your solution ensures accuracy and correctness. It involves substituting your solution back into the original equation.

For our problem, we found \(x = 3/2\) as the solution. To verify:

• Substitute \(x\) back into the original equation \((2x - 3)^6 = 0\).
• Replace \(x\) with \(3/2\): \((2(3/2) - 3)^6\).
• Simplify the expression inside the parenthesis \(2 * 3/2 - 3 = 3 - 3 = 0\).
• Finally, \(0^6 = 0\), which is true.

Verifying the solution helps avoid mistakes and confirms that the isolated variable indeed satisfies the original equation. Remember, this is a crucial final step.