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Simplifying expressions is a fundamental mathematical skill that helps to express algebraic statements in their simplest form. The process involves reducing expressions to a less complex format without changing their values. This becomes particularly useful when solving equations or working with complex mathematical statements, as it makes them easier to handle and understand.

To simplify an expression like \( (4b)^{1/2} (8b^{1/4}) \), we break it down into simpler parts. We separate the numbers from the variables and handle them individually. For example:

• The number 4 becomes \( 2 \) when raised to the power of \( 1/2 \)
• The variable \( b \) raised to a power, can also be simplified by applying its exponent rules

Combining these simplifications helps us express the problem in a cleaner and more workable form, which leads to the result \( 16b^{3/4} \), simplifying not only the numbers but also the variable terms.

Rational exponents are another way of writing roots using powers or exponents. A rational exponent indicates both a root and a power at the same time. The general format is \( a^{m/n} \), where:

• \( a \) is the base
• \( m \) is the power
• \( n \) is the root

This means that \( a^{m/n} \) is the \( n^{th} \) root of \( a \) raised to the power \( m \). For example, the expression \( (4b)^{1/2} \) uses a rational exponent to denote the square root of \( 4b \). It simplifies to \( 2b^{1/2} \), separating the function of the root and any potential power in a single expression.

Understanding rational exponents is crucial for efficiently managing and simplifying algebraic expressions that involve roots. Rather than using the radical sign, these expressions foster straightforward calculations and manipulations by treating all numbers as exponents.

Multiplying powers with the same base requires adding their exponents together. This rule stems from the repeated multiplication properties of numbers and expressions. For example, when you multiply terms like \( b^{1/2} \) and \( b^{1/4} \), you keep the base \( b \) and simply add the exponents:

• \( 1/2 + 1/4 = 3/4 \)

Thus, \( b^{1/2} \times b^{1/4} = b^{3/4} \). This principle helps in simplifying more complex expressions, turning many small operations into a much simpler one with a single exponent.

Similarly, in the expression \( 9a^{3/2} \times 5a^{1/2} \), we apply the same rule:

• Add exponents \( 3/2 + 1/2 \) resulting in \( 2 \)

This provides \( 45a^{2} \) combining both coefficients and bases. Simplifying the multiplication of powers this way not only streamlines calculations but also reveals the ends of complicated expressions with elegance and simplicity.