91Ó°ÊÓ

Inverse operations simply reverse each other. They help us isolate variables in equations. In our example, the equation was given as \( t^{-1/2} = 7 \). Here, the exponent is \( -1/2 \), meaning it's the reciprocal square root. To solve it, we need to reverse this (or the inverse operation). Let's simplify that step-by-step:
1. Reciprocal of both sides: If \( t^{-1/2} = 7 \), then \( t^{1/2} = 1/7 \).
2. We know that the inverse of square rooting a number is squaring it. So, square both sides: \((t^{1/2})^2 = (1/7)^2 \) simplifies to: \(t = 1/49\). By applying these inverse operations, we can solve for \( t \). This approach helps us understand equations better and isolate the variable we're solving for.

Exponents are used to indicate repeated multiplication of a number by itself. For example, in the term \( t^{-1/2} \), the exponent \( -1/2 \) means several things:
- The negative sign indicates a reciprocal.
- The \( 1/2 \) shows the square root.
This means \( t^{-1/2} \) is the reciprocal of the square root of \( t \). Not understanding exponents could make solving equations tough. In the solution, we switched the exponent from \( -1/2 \) to \( 1/2 \) by taking reciprocals. Then, squaring it helped us eliminate the exponent completely to solve for \( t \). Remember, knowing how to handle exponents is key to mastering algebra.

Checking your solution is crucial in mathematics to ensure correctness. After solving the equation \( t = 1/49 \), we substitute it back into the original equation to verify:
1. The original equation is \( t^{-1/2} = 7 \).
2. Substitute \( t = 1/49 \): \((1/49)^{-1/2} \).
3. Simplify this: \( 49^{1/2} \). The square root of 49 is 7.
4. Verifying: This confirms \( 7 = 7 \), showing our solution is correct.
Through checking, errors can be caught and corrected, ensuring we have the right answer. Always remember to recheck your steps by substituting the solution back. It’s a simple yet effective way to confirm accuracy.