What is another way to write x2.x4?
The Correct Answer and Explanation is :
The expression ( x^2 \cdot x^4 ) can be simplified using the laws of exponents. According to the rules of exponents, when you multiply two expressions with the same base, you can add their exponents. Therefore, the expression can be rewritten as:
[
x^2 \cdot x^4 = x^{2 + 4} = x^6
]
Explanation of Exponent Rules
- Definition of Exponents: Exponents indicate how many times a number (the base) is multiplied by itself. For example, ( x^3 ) means ( x \cdot x \cdot x ).
- Product of Powers Rule: When multiplying two powers that have the same base, you add the exponents. This rule can be expressed mathematically as:
[
a^m \cdot a^n = a^{m+n}
]
Here, ( a ) is the base, and ( m ) and ( n ) are the exponents. - Applying the Rule: In the case of ( x^2 \cdot x^4 ), both terms have the same base, which is ( x ). By applying the product of powers rule:
- Identify the exponents: ( m = 2 ) and ( n = 4 ).
- Add the exponents: ( 2 + 4 = 6 ).
- Write the result as ( x^6 ).
- Implications: This simplification is particularly useful in algebra and calculus. It allows for easier manipulation of equations and can simplify computations, especially when working with polynomial expressions.
In conclusion, ( x^2 \cdot x^4 ) simplifies to ( x^6 ) by applying the product of powers rule. This fundamental property of exponents aids in simplifying expressions and solving equations effectively. Understanding this rule is crucial for anyone working with algebra, as it forms the basis for many other mathematical concepts.