Question - Simplifying Algebraic Expression with Exponents
Solution:
The expression in the image is\[ x^2 \cdot (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \div x^2 \]First, we'll simplify the term in the denominator of the first fraction by moving it to the numerator with a negative exponent:\[ x^2 \cdot (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \cdot x^{-2} \]Now, we'll combine the terms by finding a common exponent for $$ x $$. In this case, we want to match the exponent of $$ x $$ to the smallest exponent, which is $$ -2 $$:\[ x^{2-2} \cdot (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \cdot x^{-2} \]Upon simplifying $$ x^{2-2} $$, we get:\[ (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \cdot x^{-2} \]The expression cannot be simplified further because the terms are not like terms (they have different exponents). Therefore, the simplified expression is:\[ (x^2 + 1)^{-1/2} - (x^2 + 1)^{1/2} \cdot x^{-2} \]
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