Question

Multiply. $$\left(a^{3}\right)\left(a^{6}\right)$$

Short answer

\(a^{9}\)

Step-by-step solution

Apply the Product Rule for Exponents

The Product Rule states that when multiplying two expressions with the same base, you can add the exponents. In this case, both expressions have the base of 'a', so you add the exponents: 3 + 6.

Perform the Addition

Add the exponents together to get the new exponent for the base 'a'. 3 + 6 equals 9.

Write the Final Answer

Combine the base with the new exponent to express the multiplication result: \(a^{9}\).

Key concepts

Exponentiation

Understanding how to work with exponential expressions is a fundamental skill in algebra that leads to mastery of more complex topics in mathematics. Exponentiation is a form of mathematical shorthand for expressing repeated multiplication of the same number. For instance, when you see the expression
\( a^{3} \),
it means you multiply the base, 'a,' by itself three times (
\( a \times a \times a \)). This compact form is not only convenient but also crucial as numbers and expressions become more complex.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the given exercise,
\( (a^{3})(a^{6}) \)
is an algebraic expression that involves the base 'a' raised to different powers. These expressions follow specific rules that dictate how they can be manipulated and simplified to solve algebraic problems. Understanding the properties that govern these expressions is essential to correctly solving equations, factoring, and expanding terms within the broader study of algebra.

Exponents Addition

A critical rule when dealing with exponents is the Product Rule for Exponents. This rule is applied when you multiply two expressions with the same base. Instead of multiplying the bases, you keep the base the same and add the exponents. For example, in our exercise, when multiplying
\( a^{3} \) by
\( a^{6} \), rather than performing six sets of triple multiplications, you can simply add the exponents to obtain
\( a^{3+6} = a^{9} \).
This principle not only streamlines calculations but also prevents errors in more advanced operations such as solving exponential equations or working with polynomials.

Technical Mathematics

Technical mathematics refers to the application of mathematical principles and techniques in practical contexts such as engineering, computer science, and the physical sciences. In technical fields, the Product Rule for Exponents can simplify complex calculations involving power and exponential growth. It also lays the groundwork for understanding logarithms, which are the inverses of exponents and are ubiquitous in technical and scientific calculations. Grasping these foundational concepts in technical mathematics is essential for progressing to more advanced and specialized mathematical applications.