What is x squared divided by x?
The Correct Answer and Explanation is :
The expression ( x^2 \div x ) simplifies to ( x ).
Explanation:
To understand why ( x^2 \div x = x ), let’s break down the mathematical rules at play:
- Understanding Exponents:
- An exponent tells us how many times to multiply a number by itself. For example, ( x^2 ) means ( x \times x ).
- In general, ( x^n ) (where ( n ) is a positive integer) represents ( x ) multiplied by itself ( n ) times.
- Dividing Terms with Exponents:
- When we divide terms that have the same base (like ( x )), we subtract their exponents. This is called the quotient rule for exponents, which states:
[
\frac{x^a}{x^b} = x^{a – b}
] - Here, ( a = 2 ) and ( b = 1 ), so ( x^2 \div x^1 = x^{2 – 1} = x^1 ) or simply ( x ).
- Applying the Rule to ( x^2 \div x ):
- Substitute into the rule: ( x^2 \div x = x^{2 – 1} = x ).
- Another Way to Look at It:
- You can also break down ( x^2 \div x ) as:
[
\frac{x \times x}{x}
] - Here, one ( x ) on the top cancels with the ( x ) on the bottom, leaving you with just ( x ).
- Key Insight:
- Dividing two terms with the same base simplifies the expression because of exponent subtraction. It is a shortcut for expressing repeated multiplication and allows us to simplify complex expressions efficiently.
In summary, ( x^2 \div x = x ) due to the rule for dividing exponents with the same base, where we subtract the exponents. This rule is fundamental in algebra and helps simplify expressions across various mathematical contexts.