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Evaluate 27^(2/3)

Mathematics

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Answer

B

Step-by-step solution

23 steps

Here are the solutions to the questions:

Part 1· 3 steps

Evaluate 272/327^{2/3}.

  1. Rewrite the fractional exponent.

    The denominator of the exponent indicates the root, and the numerator indicates the power.

    272/3=(273)227^{2/3} = (\sqrt[3]{27})^2

  2. Calculate the cube root of 27.

    273=3\sqrt[3]{27} = 3

  3. Square the result.

    32=93^2 = 9

    The correct option is B.

    B. 9\boxed{\text{B. } 9}

Part 2· 3 steps

Solve 2x+1=322^{x+1} = 32.

  1. Express 32 as a power of 2.

    32=2532 = 2^5

  2. Equate the bases and then the exponents.

    2x+1=252^{x+1} = 2^5 x+1=5x+1 = 5

  3. Solve for xx.

    x=5−1x = 5 - 1 x=4x = 4

    The correct option is C.

    C. x=4\boxed{\text{C. } x = 4}

Part 3· 3 steps

Evaluate log⁡101000−log⁡1010\log_{10} 1000 - \log_{10} 10.

  1. Use the logarithm property log⁡bA−log⁡bB=log⁡b(AB)\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right).

    log⁡101000−log⁡1010=log⁡10(100010)\log_{10} 1000 - \log_{10} 10 = \log_{10} \left(\frac{1000}{10}\right)

  2. Simplify the expression inside the logarithm.

    =log⁡10100= \log_{10} 100

  3. Evaluate the logarithm.

    Since 102=10010^2 = 100, log⁡10100=2\log_{10} 100 = 2.

    =2= 2

    The correct option is C.

    C. 2\boxed{\text{C. } 2}

Part 4· 2 steps

Solve log⁡3x=4\log_3 x = 4.

  1. Convert the logarithmic equation to an exponential equation.

    If log⁡bA=C\log_b A = C, then bC=Ab^C = A.

    x=34x = 3^4

  2. Calculate the value of 343^4.

    x=3×3×3×3x = 3 \times 3 \times 3 \times 3 x=81x = 81

    The correct option is D.

    D. x=81\boxed{\text{D. } x = 81}

Part 5· 2 steps

Simplify 50−8\sqrt{50} - \sqrt{8}.

  1. Simplify each square root by finding the largest perfect square factor.

    50=25×2=25×2=52\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} 8=4×2=4×2=22\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}

  2. Subtract the simplified terms.

    52−22=(5−2)25\sqrt{2} - 2\sqrt{2} = (5 - 2)\sqrt{2} =32= 3\sqrt{2}

    The correct option is D.

    D. 32\boxed{\text{D. } 3\sqrt{2}}

Part 6· 3 steps

Rationalise 6/36/\sqrt{3}.

  1. Multiply the numerator and the denominator by 3\sqrt{3} to eliminate the radical in the denominator.

    63=63×33\frac{6}{\sqrt{3}} = \frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}

  2. Perform the multiplication.

    =633= \frac{6\sqrt{3}}{3}

  3. Simplify the fraction.

    =23= 2\sqrt{3}

    The correct option is D.

    D. 23\boxed{\text{D. } 2\sqrt{3}}

Part 7· 2 steps

Solve x2−5x+6=0x^2 - 5x + 6 = 0.

  1. Factor the quadratic equation.

    Look for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3.

    (x−2)(x−3)=0(x - 2)(x - 3) = 0

  2. Set each factor equal to zero and solve for xx.

    x−2=0orx−3=0x - 2 = 0 \quad \text{or} \quad x - 3 = 0 x=2orx=3x = 2 \quad \text{or} \quad x = 3

    The correct option is B.

    B. x=2 or 3\boxed{\text{B. } x = 2 \text{ or } 3}

Part 8· 2 steps

Find the sum of the roots of 2x2−7x+3=02x^2 - 7x + 3 = 0.

  1. Identify the coefficients aa, bb, and cc from the standard quadratic equation form ax2+bx+c=0ax^2 + bx + c = 0.

    Here, a=2a = 2, b=−7b = -7, and c=3c = 3.

  2. Use the formula for the sum of the roots of a quadratic equation, which is −ba-\frac{b}{a}.

    Sum of roots=−(−7)2\text{Sum of roots} = -\frac{(-7)}{2} =72= \frac{7}{2}

    The correct option is D.

    D. 72\boxed{\text{D. } \frac{7}{2}}

Part 9· 3 steps

Find the minimum value of y=x2−6x+11y = x^2 - 6x + 11 for real xx.

  1. This is a quadratic function in the form y=ax2+bx+cy = ax^2 + bx + c.

    Since a=1>0a = 1 > 0, the parabola opens upwards, and its vertex represents the minimum point.

  2. Find the x-coordinate of the vertex using the formula x=−b2ax = -\frac{b}{2a}.

    Here, a=1a = 1 and b=−6b = -6.

    x=−(−6)2(1)x = -\frac{(-6)}{2(1)} x=62x = \frac{6}{2} x=3x = 3

  3. Substitute this x-value back into the equation to find the minimum y-value.

    y=(3)2−6(3)+11y = (3)^2 - 6(3) + 11 y=9−18+11y = 9 - 18 + 11 y=−9+11y = -9 + 11 y=2y = 2

    The correct option is B.

    B. 2\boxed{\text{B. } 2}
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