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Understanding the power rule is crucial when simplifying expressions that involve exponents. The power rule states that when you raise a power to another power, you multiply the exponents. However, when you raise a base to a fractional power, it is equivalent to taking the root of the base.

For example, if you have a fraction raised to the power of 1/2, it means taking the square root of both the numerator and the denominator. In our original exercise, the expression \[\left(\frac{9}{49}\right)^{1/2}\]indicates that each part of the fraction—both the numerator and the denominator—should be individually raised to the 1/2 power, or in other words, each should be square rooted. This understanding simplifies the process significantly and is a fundamental technique when handling fractional exponents.

Remember that the power rule is applicable beyond just taking square roots; it's a versatile tool that helps simplify many algebraic expressions by appropriately managing exponents.

Square roots simplify expressions by finding the number which, when multiplied by itself, gives the original number. It is often denoted by the exponent 1/2 in mathematical operations.

In our case, the expression is \[\left(\frac{9}{49}\right)^{1/2}\].

This operation requires you to take the square root of 9 and 49 separately. The square root of 9 is 3 because 3 multiplied by itself (3 x 3) yields 9. Similarly, the square root of 49 is 7 since 7 times 7 equals 49.

Finding the square root is a special case of exponents, where numbers are expressed as the base raised to the power of 1/2. This provides a consistent approach to simplifying expressions that resemble fractional powers in algebra. By breaking down each part of the fraction and addressing them individually, it becomes much easier to arrive at the simplified form, which here turns out to be 3/7.

When simplifying fractions involving exponents, the task often involves breaking down the number at both the numerator and the denominator levels. In this step-by-step simplification, the primary aim is to transform complex expressions into simpler forms.

Let's take the fraction \[\left(\frac{9}{49}\right)^{1/2}\] as an example. The first operation involves isolating the numerator (9) and the denominator (49) and then simplifying them separately by taking their square roots.

The numerator, 9, becomes 3 because \[9^{1/2} = 3\]. Similarly, the denominator, 49, turns into 7 since \[49^{1/2} = 7\].

• Simplify each part separately: take the root of each.
• Apply the results back to form the reduced fraction.

Once both the numerator and the denominator have been simplified, the expression should be recomposed as a fraction. This gives us the outcome: \[\frac{3}{7}\], a much simpler representation. This method is essential for efficiently handling fraction simplification with ease.