We read your photo as:

Simplify the following expressions involving surds: a) sqrt(5)(sqrt(2)+1) b) sqrt(5)(3-sqrt(2)) c) (2sqrt(2))3 d) (sqrt(5)+2)(sqrt(5)-1) e) (sqrt(7)+2)(sqrt(7)-1)

Mathematics

ScanSolve AI Answer

Step-by-step solution

37 steps

Here are the simplified expressions in surd form:

Part a· 2 steps

Simplify 5(2+1)\sqrt{5}(\sqrt{2}+1).

  1. Distribute 5\sqrt{5} to each term inside the parenthesis.

    5(2+1)=5×2+5×1\sqrt{5}(\sqrt{2}+1) = \sqrt{5} \times \sqrt{2} + \sqrt{5} \times 1

  2. Multiply the surds and simplify.

    10+5\sqrt{10} + \sqrt{5} 10+5\boxed{\sqrt{10} + \sqrt{5}}

Part b· 2 steps

Simplify 5(3−2)\sqrt{5}(3-\sqrt{2}).

  1. Distribute 5\sqrt{5} to each term inside the parenthesis.

    5(3−2)=5×3−5×2\sqrt{5}(3-\sqrt{2}) = \sqrt{5} \times 3 - \sqrt{5} \times \sqrt{2}

  2. Multiply the surds and simplify.

    35−103\sqrt{5} - \sqrt{10} 35−10\boxed{3\sqrt{5} - \sqrt{10}}

Part c· 3 steps

Simplify (22)3(2\sqrt{2})^3.

  1. Apply the exponent to both the coefficient and the surd.

    (22)3=23×(2)3(2\sqrt{2})^3 = 2^3 \times (\sqrt{2})^3

  2. Calculate the powers.

    8×(2×2×2)=8×(2×2)8 \times (\sqrt{2} \times \sqrt{2} \times \sqrt{2}) = 8 \times (2 \times \sqrt{2})

  3. Multiply the terms.

    16216\sqrt{2} 162\boxed{16\sqrt{2}}

Part d· 3 steps

Simplify (5+2)(5−1)(\sqrt{5}+2)(\sqrt{5}-1).

  1. Use the FOIL method (First, Outer, Inner, Last).

    (5+2)(5−1)=(5×5)+(5×−1)+(2×5)+(2×−1)(\sqrt{5}+2)(\sqrt{5}-1) = (\sqrt{5} \times \sqrt{5}) + (\sqrt{5} \times -1) + (2 \times \sqrt{5}) + (2 \times -1)

  2. Perform the multiplications.

    5−5+25−25 - \sqrt{5} + 2\sqrt{5} - 2

  3. Combine like terms.

    (5−2)+(−1+2)5=3+5(5-2) + (-1+2)\sqrt{5} = 3 + \sqrt{5} 3+5\boxed{3 + \sqrt{5}}

Part e· 3 steps

Simplify (7+2)(7−1)(\sqrt{7}+2)(\sqrt{7}-1).

  1. Use the FOIL method.

    (7+2)(7−1)=(7×7)+(7×−1)+(2×7)+(2×−1)(\sqrt{7}+2)(\sqrt{7}-1) = (\sqrt{7} \times \sqrt{7}) + (\sqrt{7} \times -1) + (2 \times \sqrt{7}) + (2 \times -1)

  2. Perform the multiplications.

    7−7+27−27 - \sqrt{7} + 2\sqrt{7} - 2

  3. Combine like terms.

    (7−2)+(−1+2)7=5+7(7-2) + (-1+2)\sqrt{7} = 5 + \sqrt{7} 5+7\boxed{5 + \sqrt{7}}

Part f· 3 steps

Simplify (7−6)2(\sqrt{7}-6)^2.

  1. Use the formula (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2.

    Here a=7a=\sqrt{7} and b=6b=6.

    (7−6)2=(7)2−2(7)(6)+(6)2(\sqrt{7}-6)^2 = (\sqrt{7})^2 - 2(\sqrt{7})(6) + (6)^2

  2. Perform the operations.

    7−127+367 - 12\sqrt{7} + 36

  3. Combine the constant terms.

    (7+36)−127=43−127(7+36) - 12\sqrt{7} = 43 - 12\sqrt{7} 43−127\boxed{43 - 12\sqrt{7}}

Part g· 3 steps

Simplify 915÷339\sqrt{15} \div 3\sqrt{3}.

  1. Rewrite the expression as a fraction.

    91533\frac{9\sqrt{15}}{3\sqrt{3}}

  2. Separate the coefficients and the surds.

    93×153\frac{9}{3} \times \frac{\sqrt{15}}{\sqrt{3}}

  3. Simplify the coefficients and the surds.

    3×153=353 \times \sqrt{\frac{15}{3}} = 3\sqrt{5} 35\boxed{3\sqrt{5}}

Part h· 3 steps

Simplify 730÷27\sqrt{30} \div \sqrt{2}.

  1. Rewrite the expression as a fraction.

    7302\frac{7\sqrt{30}}{\sqrt{2}}

  2. Separate the coefficient and the surds.

    7×3027 \times \frac{\sqrt{30}}{\sqrt{2}}

  3. Simplify the surds.

    7×302=7157 \times \sqrt{\frac{30}{2}} = 7\sqrt{15} 715\boxed{7\sqrt{15}}

Part h(i)· 15 steps

Simplify 111\frac{1}{\sqrt{11}}.

  1. Rationalize the denominator by multiplying the numerator and denominator by 11\sqrt{11}.

    111×1111\frac{1}{\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}}

  2. Perform the multiplication.

    1111\frac{\sqrt{11}}{11} 1111\boxed{\frac{\sqrt{11}}{11}}

    j) Simplify 303\frac{30}{\sqrt{3}}.

  3. Rationalize the denominator by multiplying the numerator and denominator by 3\sqrt{3}.

    303×33\frac{30}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}

  4. Perform the multiplication.

    3033\frac{30\sqrt{3}}{3}

  5. Simplify the fraction.

    10310\sqrt{3} 103\boxed{10\sqrt{3}}

    k) Simplify 11−3\frac{1}{1-\sqrt{3}}.

  6. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 1+31+\sqrt{3}.

    11−3×1+31+3\frac{1}{1-\sqrt{3}} \times \frac{1+\sqrt{3}}{1+\sqrt{3}}

  7. Multiply the numerators and denominators.

    Use (a−b)(a+b)=a2−b2(a-b)(a+b)=a^2-b^2 for the denominator.

    1+312−(3)2=1+31−3\frac{1+\sqrt{3}}{1^2 - (\sqrt{3})^2} = \frac{1+\sqrt{3}}{1-3}

  8. Simplify the denominator.

    1+3−2\frac{1+\sqrt{3}}{-2} −1+32\boxed{-\frac{1+\sqrt{3}}{2}}

    l) Simplify 17+2\frac{1}{\sqrt{7}+\sqrt{2}}.

  9. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 7−2\sqrt{7}-\sqrt{2}.

    17+2×7−27−2\frac{1}{\sqrt{7}+\sqrt{2}} \times \frac{\sqrt{7}-\sqrt{2}}{\sqrt{7}-\sqrt{2}}

  10. Multiply the numerators and denominators.

    Use (a+b)(a−b)=a2−b2(a+b)(a-b)=a^2-b^2 for the denominator.

    7−2(7)2−(2)2=7−27−2\frac{\sqrt{7}-\sqrt{2}}{(\sqrt{7})^2 - (\sqrt{2})^2} = \frac{\sqrt{7}-\sqrt{2}}{7-2}

  11. Simplify the denominator.

    7−25\frac{\sqrt{7}-\sqrt{2}}{5} 7−25\boxed{\frac{\sqrt{7}-\sqrt{2}}{5}}

    m) Simplify 25−7\frac{2}{\sqrt{5}-\sqrt{7}}.

  12. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 5+7\sqrt{5}+\sqrt{7}.

    25−7×5+75+7\frac{2}{\sqrt{5}-\sqrt{7}} \times \frac{\sqrt{5}+\sqrt{7}}{\sqrt{5}+\sqrt{7}}

  13. Multiply the numerators and denominators.

    Use (a−b)(a+b)=a2−b2(a-b)(a+b)=a^2-b^2 for the denominator.

    2(5+7)(5)2−(7)2=2(5+7)5−7\frac{2(\sqrt{5}+\sqrt{7})}{(\sqrt{5})^2 - (\sqrt{7})^2} = \frac{2(\sqrt{5}+\sqrt{7})}{5-7}

  14. Simplify the denominator.

    2(5+7)−2\frac{2(\sqrt{5}+\sqrt{7})}{-2}

  15. Simplify the fraction.

    −(5+7)=−5−7-(\sqrt{5}+\sqrt{7}) = -\sqrt{5}-\sqrt{7} −5−7\boxed{-\sqrt{5}-\sqrt{7}}
Was this helpful?

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

QSimplify the following Surds: a) Simplify 6/(2sqrt(2)−1). b) Simplify (2sqrt(3)+2)/(2sqrt(3)−2).QSimplify the following expressions: d) (sqrt(5)+2)(sqrt(5)−1) h) 7sqrt(30) / sqrt(2)QSimplify the following surd expressions: (i) (sqrt(5)sqrt(3) − 2sqrt(3))(sqrt(27) + 3sqrt(1))QSimplify the following expressions: e) (3)2 2 − 22 3, f) 5 × (2 + 3)2 + 13, a) 62 − 42 - sqrt(9).QSimplify the following expressions involving square roots: a) (2\[[2]5+3)([2]5+2) b)…
Browse by topic:Algebra
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.