91Ó°ÊÓ

Exponents are a way to express repeated multiplication of a number by itself. For example, the expression \(a^n\) means that the base \(a\) is multiplied by itself \(n\) times. This concept is critical because it simplifies the representation of large numbers. In our exercise, the term \(p^3\) highlights the use of exponents. Here, the variable \(p\) is raised to the power of 3, meaning \(p\) is multiplied by itself three times, i.e., \(p \cdot p \cdot p\).

Exponents follow specific rules that make calculations easier:

• Product of powers: \(a^m \times a^n = a^{m+n}\).
• Power of a power: \((a^m)^n = a^{m \cdot n}\).
• Power of a product: \((ab)^n = a^n \cdot b^n\).

These rules help simplify expressions and solve equations efficiently.

Simplifying radicals involves finding simpler terms under the radical sign without changing the value of the expression. In the problem, we see the term \(\sqrt{5p^3}\). To simplify, we try to "break down" the numbers and variables into simpler forms if possible.

For instance, if you have \(\sqrt{16}\), knowing that \(16 = 4 \times 4\), you can simplify it to \(4\). Similarly, if a variable expression is a perfect square, it can move outside the radical. Therefore, to simplify \(\sqrt{p^3}\), we note that not all parts of \(p^3\) are perfect squares. But we can express it using fractional exponents by leveraging the property of exponents that states:

• \(\sqrt{a} = a^{1/2}\)

Thus, \(\sqrt{p^3} = p^{3/2}\). This simplifies calculations and expression writing, which we will explore more in fractional exponents.

Fractional exponents provide an alternative way to express roots and powers of a number or variable. They can make it easier to work with complex expressions, including both powers and roots.

To understand fractional exponents, consider the expression \(p^{3/2}\). The denominator of the fraction (2 in this case) indicates the root, while the numerator (3) indicates the power. Thus, \(p^{3/2}\) is equivalent to the square root of \(p^3\), or \(\sqrt{p^3}\).

This representation is particularly useful as it allows one to apply the rules of exponents with ease, such as the product and power of a power rules, to expressions that include roots.

• \(p^{m/n} = \sqrt[n]{p^m}\)
• Multiplying: \(p^{a/b} \cdot p^{c/d} = p^{(ad+bc)/(bd)}\)

This flexibility makes fractional exponents a powerful tool in algebraic simplification and problem-solving.