Math Problem Statement

Express 72+334223\frac{7\sqrt2+3\sqrt3}{4\sqrt2-2\sqrt3},in term of p+qrwhere p,q abd,r,are,rational number\sqrt{r}\,\text{where }p,q\text{ abd,r,are,rational number}

Solution

We need to simplify the expression:

72+334223\frac{7\sqrt{2} + 3\sqrt{3}}{4\sqrt{2} - 2\sqrt{3}}

To express this in the form p+qrp + q\sqrt{r}, where pp, qq, and rr are rational numbers, we need to rationalize the denominator. This can be done by multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of 42234\sqrt{2} - 2\sqrt{3} is 42+234\sqrt{2} + 2\sqrt{3}. So, we multiply the numerator and denominator by this conjugate:

(72+33)(42+23)(4223)(42+23)\frac{(7\sqrt{2} + 3\sqrt{3})(4\sqrt{2} + 2\sqrt{3})}{(4\sqrt{2} - 2\sqrt{3})(4\sqrt{2} + 2\sqrt{3})}

Now, let's simplify the denominator first:

(4223)(42+23)=(42)2(23)2(4\sqrt{2} - 2\sqrt{3})(4\sqrt{2} + 2\sqrt{3}) = (4\sqrt{2})^2 - (2\sqrt{3})^2 =16×24×3=3212=20= 16 \times 2 - 4 \times 3 = 32 - 12 = 20

So, the denominator simplifies to 20.

Now, let's expand the numerator:

(72)(42)+(72)(23)+(33)(42)+(33)(23)(7\sqrt{2})(4\sqrt{2}) + (7\sqrt{2})(2\sqrt{3}) + (3\sqrt{3})(4\sqrt{2}) + (3\sqrt{3})(2\sqrt{3})

Simplifying each term:

  1. (72)(42)=28×2=56(7\sqrt{2})(4\sqrt{2}) = 28 \times 2 = 56
  2. (72)(23)=146(7\sqrt{2})(2\sqrt{3}) = 14\sqrt{6}
  3. (33)(42)=126(3\sqrt{3})(4\sqrt{2}) = 12\sqrt{6}
  4. (33)(23)=6×3=18(3\sqrt{3})(2\sqrt{3}) = 6 \times 3 = 18

Adding these together:

56+18+146+126=74+26656 + 18 + 14\sqrt{6} + 12\sqrt{6} = 74 + 26\sqrt{6}

Thus, the expression simplifies to:

74+26620\frac{74 + 26\sqrt{6}}{20}

We can now simplify by dividing both terms in the numerator by 20:

7420+26620=3710+13610\frac{74}{20} + \frac{26\sqrt{6}}{20} = \frac{37}{10} + \frac{13\sqrt{6}}{10}

So, the expression in the form p+qrp + q\sqrt{r} is:

p=3710,q=1310,r=6p = \frac{37}{10}, \, q = \frac{13}{10}, \, r = 6

The final expression is:

3710+13106\frac{37}{10} + \frac{13}{10}\sqrt{6}

Would you like any further details or clarifications?


Here are 8 related questions:

  1. How do you find the conjugate of a binomial involving square roots?
  2. What is the process for rationalizing a denominator?
  3. How do you multiply binomials involving radicals?
  4. What are the steps to simplify a radical expression?
  5. How do you combine like terms involving square roots?
  6. How can you simplify the expression a×b\sqrt{a} \times \sqrt{b}?
  7. What is the difference between rational and irrational numbers?
  8. How do you simplify expressions involving both rational and irrational components?

Tip: Always remember to look for opportunities to factor or rationalize expressions involving radicals before simplifying.

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Math Problem Analysis

Mathematical Concepts

Rationalizing Denominators
Simplifying Radical Expressions

Formulas

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Theorems

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Suitable Grade Level

High School