Write (2x)4 without exponents
The Correct Answer and Explanation is :
To write the expression ( (2x)^4 ) without exponents, we need to expand it fully.
The expression ( (2x)^4 ) means that the quantity ( 2x ) is multiplied by itself four times:
[
(2x)^4 = (2x) \cdot (2x) \cdot (2x) \cdot (2x)
]
Now, we can break this down further. Each ( 2x ) consists of a coefficient ( 2 ) and a variable ( x ). When we multiply these together, we can group the coefficients and the variables separately.
- Coefficients: The coefficient ( 2 ) appears four times. So, when we multiply the coefficients, we get:
[
2 \cdot 2 \cdot 2 \cdot 2 = 2^4 = 16
] - Variables: The variable ( x ) appears four times as well. When multiplying variables with the same base, we add their exponents:
[
x \cdot x \cdot x \cdot x = x^4
]
Putting these results together, we find that:
[
(2x)^4 = 16 \cdot x^4
]
Thus, without using exponents, we can write the expression as:
[
16x^4
]
Explanation of Exponents
Exponents are a shorthand notation that allows us to represent repeated multiplication more succinctly. For instance, ( a^n ) indicates that ( a ) is multiplied by itself ( n ) times. This is particularly useful for simplifying mathematical expressions and performing calculations, as it reduces the clutter of writing out long multiplication sequences.
In the case of ( (2x)^4 ), recognizing the structure of the expression is key. The parentheses indicate that everything inside is to be raised to the fourth power. This means we must treat both the numerical coefficient and the variable the same way. Understanding how to manipulate expressions with exponents is fundamental in algebra, as it lays the groundwork for solving equations and working with polynomials.