(root(4)(y)*sqrty)/(root(3)(y)) If the given expression is equal to y^(a) for all positive values of y, what is the value of a ?

Question

(root(4)(y)*sqrty)/(root(3)(y)) If the given expression is equal to  y^(a) for all positive values of  y, what is the value of  a ?

Bytelearn · Accepted Answer

Expressing terms with exponents: We need to express the given expression in terms of y with exponents to find the value of a. Let's start by expressing each term with exponents.Simplifying the expression: The fourth root of y is y^(1/4), and the square root of y is y^(1/2). The cube root of y is y^(1/3). So the expression becomes (y^(1/4) * y^(1/2)) / y^(1/3).Adding exponents: Using the properties of exponents, when we multiply terms with the same base, we add the exponents. So y^(1/4) * y^(1/2) becomes y^(1/4 + 1/2).Dividing terms with the same base: Adding the exponents 1/4 and 1/2, we get y^(1/4 + 2/4) which simplifies to y^(3/4).Finding a common denominator: Now we have y^(3/4) divided by y^(1/3). Using the properties of exponents, when we divide terms with the same base, we subtract the exponents. So y^(3/4) / y^(1/3) becomes y^(3/4 - 1/3).Subtracting exponents: To subtract the exponents, we need a common denominator. The common denominator of 4 and 3 is 12. So we convert 3/4 to 9/12 and 1/3 to 4/12.Final simplification: Now we subtract the exponents: 9/12 - 4/12, which equals 5/12. So y^(3/4) / y^(1/3) simplifies to y^(5/12).Value of a: Therefore, the given expression (root(4)(y)*sqrty)/(root(3)(y)) simplifies to y^(5/12). This means that a = 5/12.

$\frac{\sqrt[4]{y} \cdot \sqrt{y}}{\sqrt[3]{y}}$ $\newline$ If the given expression is equal to $y^{a}$ for all positive values of $y$ , what is the value of $a$ ?

Full solution

More problems from Find trigonometric ratios using multiple identities

Related topics

y4⋅yy3 \frac{\sqrt[4]{y} \cdot \sqrt{y}}{\sqrt[3]{y}} 3y​4y​⋅y​​\newlineIf the given expression is equal to ya y^{a} ya for all positive values of y y y, what is the value of a a a ?

$\frac{\sqrt[4]{y} \cdot \sqrt{y}}{\sqrt[3]{y}}$ $\newline$ If the given expression is equal to $y^{a}$ for all positive values of $y$ , what is the value of $a$ ?