Which of the following numbers is an irrational number? A \(sqrt(16)\) B \(sqrt(-17)\) C \(sqrt(9)\) D \(sqrt(19)\)

Step 1: Identify irrational numbers. An irrational number is a real number that cannot be expressed as a simple fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Its decimal representation is non-terminating and non-repeating.

• A: $\sqrt{16} = 4$. This is a rational number.
• B: $\sqrt{-17}$. This is an imaginary number, not a real number.
• C: $\sqrt{9} = 3$. This is a rational number.
• D: $\sqrt{19}$. Since 19 is not a perfect square, $\sqrt{19}$ is an irrational number.

The correct option is D. $\boxed{D}$

Step 1: Simplify the expressions inside the parentheses. $3(-2+6) - 2(5-4+1)$ $= 3(4) - 2(1+1)$ $= 3(4) - 2(2)$ Step 2: Perform the multiplications. $= 12 - 4$ Step 3: Perform the subtraction. $= 8$

The correct option is B. $\boxed{B}$

Step 1: Move the decimal point to the right until there is only one non-zero digit to its left. The number is $0.0000125$. Move the decimal point 5 places to the right to get $1.25$. Step 2: Determine the power of 10. Since the decimal point was moved 5 places to the right, the exponent of 10 is $-5$. $0.0000125 = 1.25 \times 10^{-5}$

The correct option is B. $\boxed{B}$

Step 1: Apply the exponent rule $x^m \times x^n = x^{m+n}$ for the base $a$. $a^3 \times a^2 c = a^{3+2} c$ Step 2: Simplify the exponent. $= a^5 c$

The correct option is A. $\boxed{A}$

Step 1: Express the base 4 as a power of 2. $4 = 2^2$ Substitute this into the expression: $\frac{4^{x+1}}{2^x} = \frac{(2^2)^{x+1}}{2^x}$ Step 2: Apply the exponent rule $(x^m)^n = x^{mn}$ to the numerator. $= \frac{2^{2(x+1)}}{2^x}$ $= \frac{2^{2x+2}}{2^x}$ Step 3: Apply the exponent rule $\frac{x^m}{x^n} = x^{m-n}$. $= 2^{(2x+2) - x}$ $= 2^{2x - x + 2}$ $= 2^{x+2}$

The correct option is C. $\boxed{C}$

Step 1: Rewrite terms with negative exponents using $x^{-n} = \frac{1}{x^n}$. $2^{-2} \times 6^3 \times 3^{-2} = \frac{1}{2^2} \times 6^3 \times \frac{1}{3^2}$ Step 2: Calculate the powers. $= \frac{1}{4} \times 216 \times \frac{1}{9}$ Step 3: Multiply the fractions. $= \frac{216}{4 \times 9}$ $= \frac{216}{36}$ Step 4: Perform the division. $= 6$

Alternatively: Step 1: Express $6^3$ as $(2 \times 3)^3$. $2^{-2} \times 6^3 \times 3^{-2} = 2^{-2} \times (2 \times 3)^3 \times 3^{-2}$ Step 2: Apply the exponent rule $(xy)^n = x^n y^n$. $= 2^{-2} \times 2^3 \times 3^3 \times 3^{-2}$ Step 3: Group terms with the same base and apply $x^m \times x^n = x^{m+n}$. $= (2^{-2} \times 2^3) \times (3^3 \times 3^{-2})$ $= 2^{-2+3} \times 3^{3-2}$ $= 2^1 \times 3^1$ Step 4: Perform the multiplication. $= 2 \times 3$ $= 6$

The correct option is A. $\boxed{A}$