Math...HELP!?!?

Question

1) Simplify the expression. Assume all variables represent nonnegative numbers.
(square root) 16a^2
2) Simplify
(64/9)^1/2
3) Simplify the following. Assume all bases are positive.
x^5/4 • x^15/4
4) Find the following root if possible.
(square root) -100
5) Rationalize the denominator.
11/ (square root) 7
6) Combine the expressions.
8x (square root) 5 - 5x (square root) 5 - 3x (square root) 5

Answer 1

1). √(16a²).

Factor 16: 16 = 4 * 4 = 4² .

√(16a²) = √(4² * a²)

= √(4a)²

= √(± 4a)²

Extract suare root.

√(16a²) = ± 4a.

Therefore, √(16a²) = ± 4a.

2). (64/9)^(1/2) .

Factor 64 : 64 = 8 * 8 = 8² .

Factor 9 : 9 = 3 * 3 = 3² .

⇒ (8²/3²)^(1/2) = ((± 8/3)²)^(1/2) .

Basic law of exponents : (a^m)⁽ⁿ⁾ = a^mn .

= (± 8/3)^{2 * 1/2}

= ± 8/3.

Therefore, (64/9)^(1/2) = ± 8/3.

3). x^5/4 * x^15/4 .

Basic law of exponents : a^m * aⁿ = a^{(m + n)} .

⇒ x^5/4 * x^15/4  = x^{(5/4 + 15/4)}

= x^{[ (5 + 15)/4 ]}

= x^20/4

= x⁵ .

Therefore, x^5/4 * x^15/4 = x⁵ .

4). √(- 100).

Factor 100 : 100 = 10 * 10 = 10² .

⇒ √(- 100) = √(- 10² ).

Substitute i² for - 1.

= √(i² * 10² )

= √(± 10i)²

Extract square root.

√(- 100) = ± 10i.

Therefore, √(- 100) = ± 10i.

5). 11/√7.

To eliminate a radicals in the a denominator, we just ratianalize the denominator.

When you rationalize the denominator, you are getting rid of the square root in the denominator. To do this, you have to multiply the top and the bottom (of the fraction) by the number in the denominator.

(11/√7) * (√7/√7) = 11√7/7.

6). 8x√5 - 5x√5 - 3x√5.

Common out √5 from the expression.

= √5(8x - 5x - 3x)

Combine like terms.

= √5(8x - 8x)

Additive inverse property : a + (- a) = 0.

= √5(0)

= 0.

Therefore, 8x√5 - 5x√5 - 3x√5 = 0.