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Understanding the properties of exponents is essential. These properties help us simplify expressions and solve equations. Here are some key properties:

• Product Rule: For any base, if you multiply terms with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).


• Quotient Rule: When dividing terms with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).


• Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).


• Zero Exponent Rule: Any base raised to the zero power is 1: \(a^0 = 1\).


• Negative Exponent Rule: A negative exponent means you take the reciprocal of the base: \(a^{-n} = \frac{1}{a^n}\).

When simplifying expressions, combining exponents is a crucial step. Let's break down part (a) from the exercise:
The expression is \(q^{\frac{2}{3}} \times q^{\frac{5}{6}}\). According to the product rule, we add the exponents since they have the same base:
First, find a common denominator for \(\frac{2}{3}\) and \(\frac{5}{6}\). The common denominator is 6.
Convert the exponents: \(\frac{2}{3} = \frac{4}{6}\) and \(\frac{5}{6}\) remains the same. Now, combine them:
\(\frac{4}{6} + \frac{5}{6}\ = \frac{9}{6}\)Simplify \(\frac{9}{6}\) to \(\frac{3}{2}\).
Thus, \(q^{\frac{2}{3}} \times q^{\frac{5}{6}}\ = q^{\frac{3}{2}}\).

Combining exponents using these properties makes our work not only easier but also more accurate.

Negative exponents can seem intimidating, but with a simple rule, they become manageable. Let's understand how they work and apply them to the exercise.
A negative exponent indicates that we should take the reciprocal of the base: \(a^{-n} = \frac{1}{a^n}\).
For instance, \(\frac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}\) from part (c) involves subtracting the exponents, as per the quotient rule:
\(n^{\frac{3}{5} - \frac{8}{5}}\) = \(n^{-\frac{5}{5}}\) = \(n^{-1}\).

According to the negative exponent rule, \(n^{-1}\) becomes \(\frac{1}{n}\).
Now the simplified expression is \(\frac{1}{n}\).

Remember, flipping the base makes working with negative exponents straightforward.

Fractional exponents are another important concept. These exponents express roots, and understanding them is key for further simplification.
Consider \((h^6)^{\frac{4}{3}}\) from part (b). The power of a power rule states that we multiply the exponents: \((a^m)^n = a^{m \times n}\).
Multiplying \(6\) by \(\frac{4}{3}\) gives:
\(6 \times \frac{4}{3} = 8\).

Thus, the simplified form is \(h^8\).

Fractional exponents can be represented as roots, such as \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). Here, the denominator indicates the root, while the numerator shows the power.
Understanding and converting fractional exponents helps simplify complex expressions accurately and efficiently.