$\sqrt{\frac{625^{-50}}{5^{-10}}}$
Answer (1)
Rewrite 625 as 5 4 .
Substitute into the expression: 5 − 10 ( 5 4 ) − 50 .
Simplify the exponent: 5 − 10 5 − 200 .
Apply the quotient rule: 5 − 190 .
Rewrite the square root as an exponent: ( 5 − 190 ) 2 1 .
Multiply the exponents: 5 − 95 .
The simplified expression is 5 − 95 .
Explanation
Understanding the Problem We are given the expression 5 − 10 62 5 − 50 . Our goal is to simplify this expression.
Rewriting 625 as a Power of 5 First, we rewrite 625 as a power of 5. Since 625 = 5 4 , we can substitute this into the expression: 5 − 10 ( 5 4 ) − 50
Simplifying the Exponent Next, we simplify the exponent in the numerator. Using the power of a power rule, ( a m ) n = a mn , we have ( 5 4 ) − 50 = 5 4 ×− 50 = 5 − 200 . So the expression becomes: 5 − 10 5 − 200
Applying the Quotient Rule Now, we use the quotient rule for exponents, which states that a n a m = a m − n . Applying this rule, we get: 5 − 200 − ( − 10 ) = 5 − 200 + 10 = 5 − 190
Rewriting the Square Root as an Exponent We can rewrite the square root as an exponent. Recall that a = a 2 1 . Therefore, we have: 5 − 190 = ( 5 − 190 ) 2 1
Multiplying the Exponents Finally, we multiply the exponents. Using the power of a power rule again, we have: 5 − 190 ⋅ 2 1 = 5 − 95 The simplified expression is 5 − 95 . If we want to express this with a positive exponent, we can rewrite it as 5 95 1 .
Final Answer Thus, the simplified expression is 5 − 95 .
Examples
Understanding exponents and roots is crucial in many scientific fields. For example, in physics, the intensity of light decreases with the square of the distance from the source. Simplifying expressions with exponents and roots helps scientists calculate and predict these changes accurately. This skill is also essential in engineering, where it's used to design structures and systems that can withstand various forces and stresses.
