Give answers for each item and each part but follow the nlsc responding, also give full notes about the topics involved basically in part b items also relevant examples

Mathematics

ScanSolve AI Answer

Answers

3 parts
a

626\sqrt{2}

b
3.146
c
2

Step-by-step solution

15 steps

Here are the answers to your questions:

Part a· 3 steps

Simplify 18+50−8\sqrt{18} + \sqrt{50} - \sqrt{8}.

  1. Simplify each surd individually by finding the largest perfect square factor.

    18=9×2=9×2=32\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} 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. Substitute the simplified surds back into the expression.

    32+52−223\sqrt{2} + 5\sqrt{2} - 2\sqrt{2}

  3. Combine the like surds.

    (3+5−2)2(3 + 5 - 2)\sqrt{2} (8−2)2(8 - 2)\sqrt{2} 62\boxed{6\sqrt{2}}

Part b3.146

Given that 2=1.414\sqrt{2} = 1.414 and 3=1.732\sqrt{3} = 1.732, evaluate 13−2\frac{1}{\sqrt{3} - \sqrt{2}} without using tables or a calculator.

  1. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

    The conjugate of 3−2\sqrt{3} - \sqrt{2} is 3+2\sqrt{3} + \sqrt{2}.

    13−2×3+23+2\frac{1}{\sqrt{3} - \sqrt{2}} \times \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}}

  2. Apply the difference of squares formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2 to the denominator.

    3+2(3)2−(2)2\frac{\sqrt{3} + \sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2} 3+23−2\frac{\sqrt{3} + \sqrt{2}}{3 - 2} 3+21=3+2\frac{\sqrt{3} + \sqrt{2}}{1} = \sqrt{3} + \sqrt{2}

  3. Substitute the given approximate values for 3\sqrt{3} and 2\sqrt{2}.

    1.732+1.4141.732 + 1.414

  4. Perform the addition.

    3.146\boxed{3.146}

Part c2

Solve for xx: 32x−1=273^{2x-1} = 27.

  1. Express both sides of the equation with the same base.

    32x−1=333^{2x-1} = 3^3

  2. Equate the exponents since the bases are the same.

    2x−1=32x - 1 = 3

  3. Solve the linear equation for xx.

    2x=3+12x = 3 + 1 2x=42x = 4 x=42x = \frac{4}{2} x=2x = \boxed{2}

    Notes on Topics Involved in Part b:

    1. Rationalizing the Denominator
    This is a key technique used to remove surds from the denominator of a fraction, making the expression simpler and easier to work with, especially for numerical evaluation.

    • Case 1: Denominator is a single surd term (e.g., a\sqrt{a}) Multiply both the numerator and the denominator by the surd in the denominator. Example: Rationalize 73\frac{7}{\sqrt{3}}.

      73×33=733\frac{7}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{7\sqrt{3}}{3}

    • Case 2: Denominator is a binomial surd (e.g., a+ba + \sqrt{b} or a+b\sqrt{a} + \sqrt{b}) Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate is formed by changing the sign between the terms. This method uses the difference of squares identity: (x+y)(x−y)=x2−y2(x+y)(x-y) = x^2 - y^2, which eliminates the surds from the denominator.

      • The conjugate of a+ba + \sqrt{b} is a−ba - \sqrt{b}.
      • The conjugate of a−b\sqrt{a} - \sqrt{b} is a+b\sqrt{a} + \sqrt{b}.

      Example: Rationalize 52+7\frac{5}{2 + \sqrt{7}}.

  4. Identify the conjugate of the denominator.

    The conjugate of $2 + \sqrt{7}$ is $2 - \sqrt{7}$.

  5. Multiply the numerator and denominator by the conjugate.

    $$
\frac{5}{2 + \sqrt{7}} \times \frac{2 - \sqrt{7}}{2 - \sqrt{7}}
$$

  6. Perform the multiplication.

    $$
\frac{5(2 - \sqrt{7})}{2^2 - (\sqrt{7})^2}
$$
$$
\frac{10 - 5\sqrt{7}}{4 - 7}
$$
$$
\frac{10 - 5\sqrt{7}}{-3} = -\frac{10 - 5\sqrt{7}}{3} \text{ or } \frac{5\sqrt{7} - 10}{3}
$$

    2. Substitution of Approximate Values
    When asked to evaluate an expression involving surds using given approximate values (e.g., 2≈1.414\sqrt{2} \approx 1.414), it is crucial to:

    • Simplify the expression completely first, including rationalizing any denominators.
    • Then, substitute the approximate values into the simplified expression. This approach minimizes calculation errors and ensures accuracy, especially when dealing with complex fractions.

    Example: Evaluate 16−5\frac{1}{\sqrt{6} - \sqrt{5}} given 6≈2.449\sqrt{6} \approx 2.449 and 5≈2.236\sqrt{5} \approx 2.236.

  7. Rationalize the denominator.

    The conjugate of 6−5\sqrt{6} - \sqrt{5} is 6+5\sqrt{6} + \sqrt{5}.

    16−5×6+56+5\frac{1}{\sqrt{6} - \sqrt{5}} \times \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} + \sqrt{5}} 6+5(6)2−(5)2\frac{\sqrt{6} + \sqrt{5}}{(\sqrt{6})^2 - (\sqrt{5})^2} 6+56−5\frac{\sqrt{6} + \sqrt{5}}{6 - 5} 6+51=6+5\frac{\sqrt{6} + \sqrt{5}}{1} = \sqrt{6} + \sqrt{5}

  8. Substitute the given approximate values.

    2.449+2.236=4.6852.449 + 2.236 = 4.685
Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QGive me answers for each but while following the nlsc responseQGive answers in English while following the nlsc uce response styleQSection B has three parts: 1, 2 and 3. Answer all the questions in Part 1. In Part 2, answer all the…QAnswer all questions in section A and three (3) from section BQThis paper is in two sections: A and B. Answer Question 1 (compulsory) in Section A and any other…
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp