Problem 6 State whether the equation is an... [FREE SOLUTION] (2024)

Short Answer

Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, \(4^3\) means we multiply 4 three times: 4 * 4 * 4. Exponents have a base (the number being multiplied) and an exponent (the number of times the base is multiplied). Using exponents makes it easier to write and understand large numbers.

Power Rule

The Power Rule is a key concept in exponent rules. It tells us how to handle exponents raised to another exponent. The rule states that \((a^m)^n = a^{m \times n}\). In the original exercise, \((5^2)^7 = 5^{14}\), we see the Power Rule at work: the exponent 2 is multiplied by 7 to get 14. Thus, the Power Rule simplifies the expression by combining the two exponents into one.

Algebraic Rules

In algebra, certain rules help us simplify and solve equations involving variables and constants. These include the Distributive Property, the Associative Property, and the Power Rule for exponents. Each rule provides a method for manipulating mathematical expressions in a consistent manner. Understanding these rules is essential for successfully solving algebraic equations and simplifying expressions.

Multiplication of Exponents

When you multiply numbers with exponents, different rules apply based on the situation. For instance, if the bases are the same, you add the exponents: \(a^m \times a^n = a^{m+n}\). For raising an exponent to another, the Power Rule applies. Also, remember that if you raise a product to a power, each factor in the product is raised to the power: \((ab)^n = a^n \times b^n\). Mastering these rules helps in working with complex algebraic expressions.