Find each product. Assume all variables represent positive real numbers. (p^1/2-p^-1/2)(p^1/2+p^-1/2)

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Recognize that the expression \((p^{1/2} - p^{-1/2})(p^{1/2} + p^{-1/2})\) is in the form of a difference of squares: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = p^{1/2}\) and \(b = p^{-1/2}\).

Apply the difference of squares formula to rewrite the product as \( (p^{1/2})^2 - (p^{-1/2})^2 \).

Simplify each term by using the property of exponents \((p^m)^n = p^{mn}\). So, \((p^{1/2})^2 = p^{(1/2) \times 2} = p^1 = p\) and \((p^{-1/2})^2 = p^{(-1/2) \times 2} = p^{-1}\).

Rewrite the expression as \(p - p^{-1}\).

Since \(p^{-1} = \frac{1}{p}\), the expression can also be written as \(p - \frac{1}{p}\), which is the simplified product.

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주요 개념

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Properties of Exponents

Understanding how to manipulate exponents is essential, including rules like multiplying powers with the same base by adding exponents and interpreting negative and fractional exponents. For example, p^(1/2) represents the square root of p, and p^(-1/2) is the reciprocal of the square root of p.

Difference of Squares Formula

The expression (a - b)(a + b) equals a^2 - b^2. Recognizing this pattern allows simplification without expanding each term individually. Here, a and b correspond to p^(1/2) and p^(-1/2), respectively.

Simplifying Algebraic Expressions

After applying exponent rules and formulas, simplifying the resulting expression by combining like terms or reducing powers is necessary. This includes rewriting expressions with fractional exponents into radical form or simpler powers.

Find each product. Assume all variables represent positive real numbers. (p1/2-p-1/2)(p1/2+p-1/2)