Which expression is equivalent to y
0
(Assume y
=0 )?
- y
0 - y 3. 0 4. 1
The Correct Answer and Explanation is :
The expression ( y^0 ) (where ( y \neq 0 )) is equivalent to 1. Thus, the correct answer is 4. 1.
Explanation
In mathematics, the notation ( y^n ) denotes ( y ) raised to the power of ( n ). When ( n ) is zero, the rule is that any non-zero number raised to the power of zero equals one. This principle is part of the laws of exponents, which govern how numbers behave when multiplied or divided in terms of powers.
Here’s why ( y^0 = 1 ):
- Definition of Exponents: By definition, the expression ( y^n ) represents ( y ) multiplied by itself ( n ) times. For example:
- ( y^3 = y \times y \times y )
- ( y^2 = y \times y )
- ( y^1 = y )
- Transitioning to Zero Power: If we look at how the powers decrease:
- ( y^3 = y \times y \times y )
- ( y^2 = y \times y )
- ( y^1 = y )
- ( y^0 = \frac{y^1}{y^1} = \frac{y}{y} = 1 ) (since ( y \neq 0 ))
This shows that ( y^0 ) is equivalent to 1 because any non-zero number divided by itself is 1.
- Consistency Across Operations: This rule helps maintain consistency in algebraic operations. For instance, when multiplying powers with the same base, we add the exponents:
[
y^a \times y^b = y^{a+b}
]
If ( b = 0 ), then ( y^a \times y^0 = y^{a+0} = y^a ). To maintain equality, ( y^0 ) must equal 1. - Applications: Understanding this principle is crucial in algebra and higher mathematics, where manipulating expressions often involves changing powers and simplifying equations.
In conclusion, ( y^0 ) equals 1 when ( y ) is any non-zero number, solidifying the result that ( y^0 = 1 ) is universally accepted in mathematics.