The expression $\frac{y^a y^b \sqrt{y}}{\sqrt[5]{y^7}}$, where $a=\frac{3}{4}$ and $b=-\frac{5}{8}$, is equivalent to $y^c$. What is the value of $c$?
Answer (2)
Rewrite the expression using exponents: 5 y 7 y a y b y = y 5 7 y a y b y 2 1 .
Simplify the numerator using the property y m y n = y m + n : y a y b y 2 1 = y a + b + 2 1 = y 4 3 − 8 5 + 2 1 = y 8 5 .
Simplify the entire expression using the property y n y m = y m − n : y 5 7 y 8 5 = y 8 5 − 5 7 = y − 40 31 .
Therefore, c = − 40 31 .
Explanation
Understanding the Problem We are given the expression 5 y 7 y a y b y , where a = 4 3 and b = − 8 5 , and we want to find the value of c such that this expression is equivalent to y c . In other words, we want to simplify the given expression and express it in the form y c .
Rewriting with Exponents First, let's rewrite the expression using exponents. Recall that y = y 2 1 and 5 y 7 = y 5 7 . So, the expression becomes 5 y 7 y a y b y = y 5 7 y a y b y 2 1 .
Simplifying the Numerator Next, we use the property of exponents that y m y n = y m + n . Thus, the numerator becomes y a y b y 2 1 = y a + b + 2 1 .
Calculating the Numerator's Exponent Now, we substitute the given values of a and b into the exponent: a + b + 2 1 = 4 3 + ( − 8 5 ) + 2 1 = 4 3 − 8 5 + 2 1 . To add these fractions, we need a common denominator, which is 8. So we have 8 6 − 8 5 + 8 4 = 8 6 − 5 + 4 = 8 5 . Therefore, the numerator simplifies to y 8 5 .
Simplifying the Fraction Now our expression looks like this: y 5 7 y 8 5 . To simplify this fraction, we use the property of exponents that y n y m = y m − n . So we have y 5 7 y 8 5 = y 8 5 − 5 7 .
Calculating the Final Exponent Now we need to calculate the exponent 8 5 − 5 7 . To subtract these fractions, we need a common denominator, which is 40. So we have 8 5 − 5 7 = 40 25 − 40 56 = 40 25 − 56 = 40 − 31 = − 40 31 .
Finding the Value of c Therefore, the expression simplifies to y − 40 31 . Since we are given that the expression is equivalent to y c , we have c = − 40 31 .
Final Answer Thus, the value of c is − 40 31 .
Examples
Understanding exponents and their properties is crucial in many scientific fields, such as physics and computer science. For example, in physics, the intensity of light decreases with the square of the distance from the source. This relationship can be expressed using exponents. In computer science, exponents are used to describe the growth rate of algorithms. Simplifying expressions with exponents, as we did in this problem, allows scientists and engineers to make calculations and predictions about the behavior of various systems.
The value of c in the expression 5 y 7 y a y b y is − 40 31 after simplification. The process includes rewriting components, simplifying the numerator and using properties of exponents to resolve the expression. Ultimately, this results in y c where c = − 40 31 .
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