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

Mathematics

ScanSolve AI Answer

Answer

D

Step-by-step solution

19 steps

  1. Identify irrational numbers.

    An irrational number is a real number that cannot be expressed as a simple fraction pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0. Its decimal representation is non-terminating and non-repeating.

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

    The correct option is D.

    D\boxed{\text{D}}

  2. Simplify the expressions inside the parentheses.

    3(−2+6)−2(5−4+1)3(-2+6) - 2(5-4+1) =3(4)−2(1+1)= 3(4) - 2(1+1) =3(4)−2(2)= 3(4) - 2(2)

  3. Perform the multiplications.

    =12−4= 12 - 4

  4. Perform the subtraction.

    =8= 8

    The correct option is B.

    B\boxed{\text{B}}

  5. 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.

  6. Determine the power of 10.

    Since the decimal point was moved 5 places to the right, the exponent of 10 is −5-5.

    0.0000125=1.25×10−50.0000125 = 1.25 \times 10^{-5}

    The correct option is B.

    B\boxed{\text{B}}

  7. Apply the exponent rule xm×xn=xm+nx^m \times x^n = x^{m+n} for the base aa.

    a3×a2c=a3+2ca^3 \times a^2 c = a^{3+2} c

  8. Simplify the exponent.

    =a5c= a^5 c

    The correct option is A.

    A\boxed{\text{A}}

  9. Express the base 4 as a power of 2.

    4=224 = 2^2

    Substitute this into the expression:

    4x+12x=(22)x+12x\frac{4^{x+1}}{2^x} = \frac{(2^2)^{x+1}}{2^x}

  10. Apply the exponent rule (xm)n=xmn(x^m)^n = x^{mn} to the numerator.

    =22(x+1)2x= \frac{2^{2(x+1)}}{2^x} =22x+22x= \frac{2^{2x+2}}{2^x}

  11. Apply the exponent rule xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}.

    =2(2x+2)−x= 2^{(2x+2) - x} =22x−x+2= 2^{2x - x + 2} =2x+2= 2^{x+2}

    The correct option is C.

    C\boxed{\text{C}}

  12. Rewrite terms with negative exponents using x−n=1xnx^{-n} = \frac{1}{x^n}.

    2−2×63×3−2=122×63×1322^{-2} \times 6^3 \times 3^{-2} = \frac{1}{2^2} \times 6^3 \times \frac{1}{3^2}

  13. Calculate the powers.

    =14×216×19= \frac{1}{4} \times 216 \times \frac{1}{9}

  14. Multiply the fractions.

    =2164×9= \frac{216}{4 \times 9} =21636= \frac{216}{36}

  15. Perform the division.

    =6= 6

    Alternatively:

  16. Express 636^3 as (2×3)3(2 \times 3)^3.

    2−2×63×3−2=2−2×(2×3)3×3−22^{-2} \times 6^3 \times 3^{-2} = 2^{-2} \times (2 \times 3)^3 \times 3^{-2}

  17. Apply the exponent rule (xy)n=xnyn(xy)^n = x^n y^n.

    =2−2×23×33×3−2= 2^{-2} \times 2^3 \times 3^3 \times 3^{-2}

  18. Group terms with the same base and apply xm×xn=xm+nx^m \times x^n = x^{m+n}.

    =(2−2×23)×(33×3−2)= (2^{-2} \times 2^3) \times (3^3 \times 3^{-2}) =2−2+3×33−2= 2^{-2+3} \times 3^{3-2} =21×31= 2^1 \times 3^1

  19. Perform the multiplication.

    =2×3= 2 \times 3 =6= 6

    The correct option is A.

    A\boxed{\text{A}}
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