Express as a single fraction: 2 rac12 + 3 rac14 rdiv 2 rac16 - 1 rac16

Question

Express as a single fraction: 2 rac12 + 3 rac14 rdiv 2 rac16 - 1 rac16

ScanSolve AI Tutor · Accepted Answer

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2. Express as a single fraction: $2\frac{1}{2} + 3\frac{1}{4} \div 2\frac{1}{6} - 1\frac{1}{6}$

Convert all mixed numbers to improper fractions.
$2\frac{1}{2} = \frac{5}{2}$ $3\frac{1}{4} = \frac{13}{4}$ $2\frac{1}{6} = \frac{13}{6}$ $1\frac{1}{6} = \frac{7}{6}$
The expression becomes:
$\frac{5}{2} + \frac{13}{4} \div \frac{13}{6} - \frac{7}{6}$
Perform the division first, following the order of operations.
$\frac{13}{4} \div \frac{13}{6} = \frac{13}{4} \times \frac{6}{13} = \frac{6}{4} = \frac{3}{2}$
The expression is now:
$\frac{5}{2} + \frac{3}{2} - \frac{7}{6}$
Perform the addition and subtraction from left to right.
$\frac{5}{2} + \frac{3}{2} = \frac{5+3}{2} = \frac{8}{2} = 4$
The expression is now:
$4 - \frac{7}{6}$
Subtract the fraction.
$4 - \frac{7}{6} = \frac{24}{6} - \frac{7}{6} = \frac{24-7}{6} = \frac{17}{6}$
The final answer is:
$\boxed{\frac{17}{6}}$
3. By rationalizing the denominator, express $A = \frac{3-\sqrt{2}}{1+\sqrt{2}}$ into the form $a + b\sqrt{2}$ where $a$ and $b$ are constants.
Identify the conjugate of the denominator.
The denominator is $1+\sqrt{2}$ . Its conjugate is $1-\sqrt{2}$ .
Multiply the numerator and denominator by the conjugate.
$A = \frac{3-\sqrt{2}}{1+\sqrt{2}} \times \frac{1-\sqrt{2}}{1-\sqrt{2}}$
Expand the numerator and denominator.
Numerator: $(3-\sqrt{2})(1-\sqrt{2}) = 3(1) + 3(-\sqrt{2}) - \sqrt{2}(1) - \sqrt{2}(-\sqrt{2})$
$= 3 - 3\sqrt{2} - \sqrt{2} + (\sqrt{2})^2$ $= 3 - 4\sqrt{2} + 2$ $= 5 - 4\sqrt{2}$
Denominator: $(1+\sqrt{2})(1-\sqrt{2}) = 1^2 - (\sqrt{2})^2$
$= 1 - 2$ $= -1$
Combine the simplified numerator and denominator and express in the form $a+b\sqrt{2}$ .
$A = \frac{5 - 4\sqrt{2}}{-1}$ $A = -5 + 4\sqrt{2}$
Comparing this to $a+b\sqrt{2}$ , we have $a=-5$ and $b=4$ .
The final answer is:
$\boxed{-5 + 4\sqrt{2}}$

Express as a single fraction: 2 rac12 + 3 rac14 rdiv 2 rac16 - 1 rac16

Step-by-step solution

Convert all mixed numbers to improper fractions.

Perform the division first, following the order of operations.

Perform the addition and subtraction from left to right.

Subtract the fraction.

Identify the conjugate of the denominator.

Multiply the numerator and denominator by the conjugate.

Expand the numerator and denominator.

Combine the simplified numerator and denominator and express in the form $a+b\sqrt{2}$ .

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More Mathematics Questions

Express as a single fraction: 2 rac12 + 3 rac14 rdiv 2 rac16 - 1 rac16

Step-by-step solution

Convert all mixed numbers to improper fractions.

Perform the division first, following the order of operations.

Perform the addition and subtraction from left to right.

Subtract the fraction.

Identify the conjugate of the denominator.

Multiply the numerator and denominator by the conjugate.

Expand the numerator and denominator.

Combine the simplified numerator and denominator and express in the form a+b2a+b\sqrt{2}a+b2​.

Still stuck on this one?

More Mathematics Questions

Express as a single fraction: 2 rac12 + 3 rac14 rdiv 2 rac16 - 1 rac16

Combine the simplified numerator and denominator and express in the form $a+b\sqrt{2}$ .