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)

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Asked on April 19, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-19T09:28:33.056Z

You're on a roll — Here are the simplified expressions in surd form: a) Simplify sqrt(5)(sqrt(2)+1). Step 1: Distribute sqrt(5) to each term inside the parenthesis. sqrt(5)(sqrt(2)+1) = sqrt(5) × sqrt(2) + sqrt(5) × 1 Step 2: Multiply the surds and simplify. sqrt(10) + sqrt(5) sqrt(10) + sqrt(5) b) Simplify sqrt(5)(3-sqrt(2)). Step 1: Distribute sqrt(5) to each term inside the parenthesis. sqrt(5)(3-sqrt(2)) = sqrt(5) × 3 - sqrt(5) × sqrt(2) Step 2: Multiply the surds and simplify. 3sqrt(5) - sqrt(10) 3sqrt(5) - sqrt(10) c) Simplify (2sqrt(2))^3. Step 1: Apply the exponent to both the coefficient and the surd. (2sqrt(2))^3 = 2^3 × (sqrt(2))^3 Step 2: Calculate the powers. 8 × (sqrt(2) × sqrt(2) × sqrt(2)) = 8 × (2 × sqrt(2)) Step 3: Multiply the terms. 16sqrt(2) 16sqrt(2) d) Simplify (sqrt(5)+2)(sqrt(5)-1). Step 1: Use the FOIL method (First, Outer, Inner, Last). (sqrt(5)+2)(sqrt(5)-1) = (sqrt(5) × sqrt(5)) + (sqrt(5) × -1) + (2 × sqrt(5)) + (2 × -1) Step 2: Perform the multiplications. 5 - sqrt(5) + 2sqrt(5) - 2 Step 3: Combine like terms. (5-2) + (-1+2)sqrt(5) = 3 + sqrt(5) 3 + sqrt(5) e) Simplify (sqrt(7)+2)(sqrt(7)-1). Step 1: Use the FOIL method. (sqrt(7)+2)(sqrt(7)-1) = (sqrt(7) × sqrt(7)) + (sqrt(7) × -1) + (2 × sqrt(7)) + (2 × -1) Step 2: Perform the multiplications. 7 - sqrt(7) + 2sqrt(7) - 2 Step 3: Combine like terms. (7-2) + (-1+2)sqrt(7) = 5 + sqrt(7) 5 + sqrt(7) f) Simplify (sqrt(7)-6)^2. Step 1: Use the formula (a-b)^2 = a^2 - 2ab + b^2. Here a=sqrt(7) and b=6. (sqrt(7)-6)^2 = (sqrt(7))^2 - 2(sqrt(7))(6) + (6)^2 Step 2: Perform the operations. 7 - 12sqrt(7) + 36 Step 3: Combine the constant terms. (7+36) - 12sqrt(7) = 43 - 12sqrt(7) 43 - 12sqrt(7) g) Simplify 9sqrt(15) ÷ 3sqrt(3). Step 1: Rewrite the expression as a fraction. 9sqrt(15)3sqrt(3) Step 2: Separate the coefficients and the surds. (9)/(3) × sqrt(15)sqrt(3) Step 3: Simplify the coefficients and the surds. 3 × sqrt((15)/(3)) = 3sqrt(5) 3sqrt(5) h) Simplify 7sqrt(30) ÷ sqrt(2). Step 1: Rewrite the expression as a fraction. 7sqrt(30)sqrt(2) Step 2: Separate the coefficient and the surds. 7 × sqrt(30)sqrt(2) Step 3: Simplify the surds. 7 × sqrt((30)/(2)) = 7sqrt(15) 7sqrt(15) i) Simplify (1)/(sqrt(11)). Step 1: Rationalize the denominator by multiplying the numerator and denominator by sqrt(11). (1)/(sqrt(11)) × sqrt(11)sqrt(11) Step 2: Perform the multiplication. sqrt(11)11 sqrt(11)11 j) Simplify (30)/(sqrt(3)). Step 1: Rationalize the denominator by multiplying the numerator and denominator by sqrt(3). (30)/(sqrt(3)) × sqrt(3)sqrt(3) Step 2: Perform the multiplication. 30sqrt(3)3 Step 3: Simplify the fraction. 10sqrt(3) 10sqrt(3) k) Simplify (1)/(1-sqrt(3)). Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 1+sqrt(3). (1)/(1-sqrt(3)) × 1+sqrt(3)1+sqrt(3) Step 2: Multiply the numerators and denominators. Use (a-b)(a+b)=a^2-b^2 for the denominator. 1+sqrt(3)1^2 - (sqrt(3))^2 = 1+sqrt(3)1-3 Step 3: Simplify the denominator. 1+sqrt(3)-2 -1+sqrt(3)2 l) Simplify (1)/(sqrt(7)+2). Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is sqrt(7)-sqrt(2). (1)/(sqrt(7)+2) × sqrt(7)-sqrt(2)sqrt(7)-sqrt(2) Step 2: Multiply the numerators and denominators. Use (a+b)(a-b)=a^2-b^2 for the denominator. sqrt(7)-sqrt(2)(sqrt(7))^2 - (sqrt(2))^2 = sqrt(7)-sqrt(2)7-2 Step 3: Simplify the denominator. sqrt(7)-sqrt(2)5 sqrt(7)-sqrt(2)5 m) Simplify (2)/(sqrt(5)-7). Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is sqrt(5)+sqrt(7). (2)/(sqrt(5)-7) × sqrt(5)+sqrt(7)sqrt(5)+sqrt(7) Step 2: Multiply the numerators and denominators. Use (a-b)(a+b)=a^2-b^2 for the denominator. 2(sqrt(5)+sqrt(7))(sqrt(5))^2 - (sqrt(7))^2 = 2(sqrt(5)+sqrt(7))5-7 Step 3: Simplify the denominator. 2(sqrt(5)+sqrt(7))-2 Step 4: Simplify the fraction. -(sqrt(5)+sqrt(7)) = -sqrt(5)-sqrt(7) -sqrt(5)-sqrt(7) Send me the next one 📸

Accepted Answer · 2026-04-19T09:28:33.056Z

You're on a roll — Here are the simplified expressions in surd form: a) Simplify sqrt(5)(sqrt(2)+1). Step 1: Distribute sqrt(5) to each term inside the parenthesis. sqrt(5)(sqrt(2)+1) = sqrt(5) × sqrt(2) + sqrt(5) × 1 Step 2: Multiply the surds and simplify. sqrt(10) + sqrt(5) sqrt(10) + sqrt(5) b) Simplify sqrt(5)(3-sqrt(2)). Step 1: Distribute sqrt(5) to each term inside the parenthesis. sqrt(5)(3-sqrt(2)) = sqrt(5) × 3 - sqrt(5) × sqrt(2) Step 2: Multiply the surds and simplify. 3sqrt(5) - sqrt(10) 3sqrt(5) - sqrt(10) c) Simplify (2sqrt(2))^3. Step 1: Apply the exponent to both the coefficient and the surd. (2sqrt(2))^3 = 2^3 × (sqrt(2))^3 Step 2: Calculate the powers. 8 × (sqrt(2) × sqrt(2) × sqrt(2)) = 8 × (2 × sqrt(2)) Step 3: Multiply the terms. 16sqrt(2) 16sqrt(2) d) Simplify (sqrt(5)+2)(sqrt(5)-1). Step 1: Use the FOIL method (First, Outer, Inner, Last). (sqrt(5)+2)(sqrt(5)-1) = (sqrt(5) × sqrt(5)) + (sqrt(5) × -1) + (2 × sqrt(5)) + (2 × -1) Step 2: Perform the multiplications. 5 - sqrt(5) + 2sqrt(5) - 2 Step 3: Combine like terms. (5-2) + (-1+2)sqrt(5) = 3 + sqrt(5) 3 + sqrt(5) e) Simplify (sqrt(7)+2)(sqrt(7)-1). Step 1: Use the FOIL method. (sqrt(7)+2)(sqrt(7)-1) = (sqrt(7) × sqrt(7)) + (sqrt(7) × -1) + (2 × sqrt(7)) + (2 × -1) Step 2: Perform the multiplications. 7 - sqrt(7) + 2sqrt(7) - 2 Step 3: Combine like terms. (7-2) + (-1+2)sqrt(7) = 5 + sqrt(7) 5 + sqrt(7) f) Simplify (sqrt(7)-6)^2. Step 1: Use the formula (a-b)^2 = a^2 - 2ab + b^2. Here a=sqrt(7) and b=6. (sqrt(7)-6)^2 = (sqrt(7))^2 - 2(sqrt(7))(6) + (6)^2 Step 2: Perform the operations. 7 - 12sqrt(7) + 36 Step 3: Combine the constant terms. (7+36) - 12sqrt(7) = 43 - 12sqrt(7) 43 - 12sqrt(7) g) Simplify 9sqrt(15) ÷ 3sqrt(3). Step 1: Rewrite the expression as a fraction. 9sqrt(15)3sqrt(3) Step 2: Separate the coefficients and the surds. (9)/(3) × sqrt(15)sqrt(3) Step 3: Simplify the coefficients and the surds. 3 × sqrt((15)/(3)) = 3sqrt(5) 3sqrt(5) h) Simplify 7sqrt(30) ÷ sqrt(2). Step 1: Rewrite the expression as a fraction. 7sqrt(30)sqrt(2) Step 2: Separate the coefficient and the surds. 7 × sqrt(30)sqrt(2) Step 3: Simplify the surds. 7 × sqrt((30)/(2)) = 7sqrt(15) 7sqrt(15) i) Simplify (1)/(sqrt(11)). Step 1: Rationalize the denominator by multiplying the numerator and denominator by sqrt(11). (1)/(sqrt(11)) × sqrt(11)sqrt(11) Step 2: Perform the multiplication. sqrt(11)11 sqrt(11)11 j) Simplify (30)/(sqrt(3)). Step 1: Rationalize the denominator by multiplying the numerator and denominator by sqrt(3). (30)/(sqrt(3)) × sqrt(3)sqrt(3) Step 2: Perform the multiplication. 30sqrt(3)3 Step 3: Simplify the fraction. 10sqrt(3) 10sqrt(3) k) Simplify (1)/(1-sqrt(3)). Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 1+sqrt(3). (1)/(1-sqrt(3)) × 1+sqrt(3)1+sqrt(3) Step 2: Multiply the numerators and denominators. Use (a-b)(a+b)=a^2-b^2 for the denominator. 1+sqrt(3)1^2 - (sqrt(3))^2 = 1+sqrt(3)1-3 Step 3: Simplify the denominator. 1+sqrt(3)-2 -1+sqrt(3)2 l) Simplify (1)/(sqrt(7)+2). Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is sqrt(7)-sqrt(2). (1)/(sqrt(7)+2) × sqrt(7)-sqrt(2)sqrt(7)-sqrt(2) Step 2: Multiply the numerators and denominators. Use (a+b)(a-b)=a^2-b^2 for the denominator. sqrt(7)-sqrt(2)(sqrt(7))^2 - (sqrt(2))^2 = sqrt(7)-sqrt(2)7-2 Step 3: Simplify the denominator. sqrt(7)-sqrt(2)5 sqrt(7)-sqrt(2)5 m) Simplify (2)/(sqrt(5)-7). Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is sqrt(5)+sqrt(7). (2)/(sqrt(5)-7) × sqrt(5)+sqrt(7)sqrt(5)+sqrt(7) Step 2: Multiply the numerators and denominators. Use (a-b)(a+b)=a^2-b^2 for the denominator. 2(sqrt(5)+sqrt(7))(sqrt(5))^2 - (sqrt(7))^2 = 2(sqrt(5)+sqrt(7))5-7 Step 3: Simplify the denominator. 2(sqrt(5)+sqrt(7))-2 Step 4: Simplify the fraction. -(sqrt(5)+sqrt(7)) = -sqrt(5)-sqrt(7) -sqrt(5)-sqrt(7) Send me the next one 📸

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)

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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)

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