5. What is the inverse of f if f(x) = ^3√x-7
8. Multiply and simplify if possible. √5x(√x-5√5)
10. 7^1/3 * 7^2/3
#5 (x+7)^3
#8 √(5x)(√x-5√5) = x√5 - 25√x
(√5)x(√x-5√5) = x√(5x) - 25x
not sure either one is any simpler
#10 add exponents: 7^(1/3 + 2/3) = 7^1 = 7
#5 might also be x^3+7
depending in just what is in the cube root
5. To find the inverse of a function f, you need to switch the roles of x and y and solve for y. Start with the function f(x) = ^3√x-7.
Step 1: Replace f(x) with y: y = ^3√x-7.
Step 2: Swap x and y: x = ^3√y-7.
Step 3: Solve for y: Cube both sides of the equation to eliminate the cube root: x^3 = y - 7.
Step 4: Isolate y: Add 7 to both sides: x^3 + 7 = y.
Step 5: Replace y with f^(-1)(x): f^(-1)(x) = x^3 + 7.
Therefore, the inverse of f(x) = ^3√x-7 is f^(-1)(x) = x^3 + 7.
8. To multiply and simplify the expression √5x(√x-5√5), you can apply the distributive property of multiplication.
Step 1: Distribute the √5x to both terms inside the parentheses: √5x * √x - √5x * 5√5.
Step 2: Simplify each term separately: √(5x) * √(x) - 5√(5x) * √(5).
Step 3: Multiply the square roots: √(5x * x) - 5√(5x * 5).
Step 4: Simplify the products inside the square roots: √(5x^2) - 5√(25x).
Step 5: Further simplify: √(5x^2) - 5√(5^2 * x).
Step 6: Simplify the square root of a square: |5x| - 5 * 5√x.
Step 7: Simplify: 5x - 25√x.
Therefore, the simplified expression of √5x(√x-5√5) is 5x - 25√x.
10. To multiply numbers with the same base, but different exponents, you can use the property of exponents that states: a^b * a^c = a^(b+c).
In this case, we have 7^(1/3) * 7^(2/3).
Using the above property, add the exponents together: 1/3 + 2/3 = 3/3 = 1.
Therefore, 7^(1/3) * 7^(2/3) simplifies to 7^1, which is just 7.