what is the simplified form of (81x^4y^9)^1/2

C 9 *x^2 y^ 4 y^ 1/2

Step-by-step explanation:

(81x^4y^9)^1/2

We know that (ab)^1/2 = a^1/2  *b^1/2

(81) ^1/2 * (x^4) ^1/2  * (y^9)^1/2

We know that a^b^c = a^ (b*c)

(81) ^1/2 * (x^(4*1/2)   * (y^(9*1/2)

9 *x^2 y^ 9/2

We know that y^9/2 = y^4 y^1/2

9 *x^2 y^ 4 y^ 1/2

A point 5.103km from B

Step-by-step explanation:

Let x be the distance from B to C. Thus, the distance from the island to C is given as;

√(x²+5²) = √(x²+25)

Mow, the distance from C to D will be equal to 13-x

Hence the total distance travelled will now be;

√(x²+25) + 13-x

Let's say k is the energy per km it takes to fly over land. Thus, total energy is given as;

E(x) = 1.4k√(x²+25) + k(13-x)

Mow, let's find the derivative of E(x);

E'(x) = 1.4k(1/2)(2x)[(x²+25)^(-1/2)] + k(-1)

Thus;

E'(x) = 1.4kx/(√(x²+25)) - k

Simplifying this to get;

E'(x) = [1.4kx - k(√(x²+25))]/[√(x²+25)]

Now, to find the distance, it will be at E'(x) = 0

Thus;

[1.4kx - k(√(x²+25))]/[√(x²+25)] = 0

Multiply both sides by [√(x²+25)] to obtain;

[1.4kx - k(√(x²+25))] = 0

(divide each term by k) to obtain ;

1.4x - (√(x²+25)) = 0

1.4x = (√(x²+25))

Square both sides;

1.4²x² = x²+25

1.96x² = x² + 25

1.96x² - x² = 25

0.96 x² = 25

x² = 25/0.96

x² = 26.042

x = √26.042

x = 5.103km